code: def _solve(f, *symbols, **flags):
      """Return a checked solution for *f* in terms of one or more of the
      symbols in the form of a list of dictionaries.
  
      If no method is implemented to solve the equation, a NotImplementedError
      will be raised. In the case that conversion of an expression to a Poly
      gives None a ValueError will be raised.
      """
  
      not_impl_msg = (
          "No algorithms are implemented to solve equation %s"
      )
  
      if len(symbols) != 1:
          # look for solutions for desired symbols that are independent
          # of symbols already solved for, e.g. if we solve for x = y
          # then no symbol having x in its solution will be returned.
  
          # First solve for linear symbols (since that is easier and limits
          # solution size) and then proceed with symbols appearing
          # in a non-linear fashion. Ideally, if one is solving a single
          # expression for several symbols, they would have to be
          # appear in factors of an expression, but we do not here
          # attempt factorization.  XXX perhaps handling a Mul
          # should come first in this routine whether there is
          # one or several symbols.
          nonlin_s = []
          got_s = set()
          rhs_s = set()
          result = []
          for s in symbols:
              xi, v = solve_linear(f, symbols=[s])
              if xi == s:
                  # no need to check but we should simplify if desired
                  if flags.get("simplify", True):
                      v = simplify(v)
                  vfree = v.free_symbols
                  if vfree & got_s:
                      # was linear, but has redundant relationship
                      # e.g. x - y = 0 has y == x is redundant for x == y
                      # so ignore
                      continue
                  rhs_s |= vfree
                  got_s.add(xi)
                  result.append({xi: v})
              elif (
                  xi
              ):  # there might be a non-linear solution if xi is not 0
                  nonlin_s.append(s)
          if not nonlin_s:
              return result
          for s in nonlin_s:
              try:
                  soln = _solve(f, s, **flags)
                  for sol in soln:
                      if sol[s].free_symbols & got_s:
                          # depends on previously solved symbols: ignore
                          continue
                      got_s.add(s)
                      result.append(sol)
              except NotImplementedError:
                  continue
          if got_s:
              return result
          else:
              raise NotImplementedError(not_impl_msg % f)
  
      # solve f for a single variable
  
      symbol = symbols[0]
  
      # expand binomials only if it has the unknown symbol
      f = f.replace(
          lambda e: isinstance(e, binomial) and e.has(symbol),
          lambda e: expand_func(e),
      )
  
      # checking will be done unless it is turned off before making a
      # recursive call; the variables `checkdens` and `check` are
      # captured here (for reference below) in case flag value changes
      flags["check"] = checkdens = check = flags.pop("check", True)
  
      # build up solutions if f is a Mul
      if f.is_Mul:
          result = set()
          for m in f.args:
              if m in {
                  S.NegativeInfinity,
                  S.ComplexInfinity,
                  S.Infinity,
              }:
                  result = set()
                  break
              soln = _vsolve(m, symbol, **flags)
              result.update(set(soln))
          result = [{symbol: v} for v in result]
          if check:
              # all solutions have been checked but now we must
              # check that the solutions do not set denominators
              # in any factor to zero
              dens = flags.get(
                  "_denominators", _simple_dens(f, symbols)
              )
              result = [
                  s
                  for s in result
                  if not any(checksol(den, s, **flags) for den in dens)
              ]
          # set flags for quick exit at end; solutions for each
          # factor were already checked and simplified
          check = False
          flags["simplify"] = False
  
      elif f.is_Piecewise:
          result = set()
          if any(e.is_zero for e, c in f.args):
              f = f.simplify()  # failure imminent w/o help
          for i, (expr, cond) in enumerate(f.args):
              if expr.is_zero:
                  raise NotImplementedError(
                      filldedent(
                          """
                      An expression is already zero when %s.
                      This means that in this *region* the solution
                      is zero but solve can only represent discrete,
                      not interval, solutions. If this is a spurious
                      interval it might be resolved with simplification
                      of the Piecewise conditions."""
                          % cond
                      )
                  )
              candidates = _vsolve(expr, symbol, **flags)
              # the explicit condition for this expr is the current cond
              # and none of the previous conditions
              args = [~c for _, c in f.args[:i]] + [cond]
              cond = And(*args)
              for candidate in candidates:
                  if candidate in result:
                      # an unconditional value was already there
                      continue
                  try:
                      v = cond.subs(symbol, candidate)
                      _eval_simplify = getattr(
                          v, "_eval_simplify", None
                      )
                      if _eval_simplify is not None:
                          # unconditionally take the simplification of v
                          v = _eval_simplify(
                              ratio=2, measure=lambda x: 1
                          )
                  except TypeError:
                      # incompatible type with condition(s)
                      continue
                  if v == False:
                      continue
                  if v == True:
                      result.add(candidate)
                  else:
                      result.add(
                          Piecewise((candidate, v), (S.NaN, True))
                      )
          # solutions already checked and simplified
          # ****************************************
          return [{symbol: r} for r in result]
      else:
          # first see if it really depends on symbol and whether there
          # is only a linear solution
          f_num, sol = solve_linear(f, symbols=symbols)
          if f_num.is_zero or sol is S.NaN:
              return []
          elif f_num.is_Symbol:
              # no need to check but simplify if desired
              if flags.get("simplify", True):
                  sol = simplify(sol)
              return [{f_num: sol}]
  
          poly = None
          # check for a single Add generator
          if not f_num.is_Add:
              add_args = [
                  i
                  for i in f_num.atoms(Add)
                  if symbol in i.free_symbols
              ]
              if len(add_args) == 1:
                  gen = add_args[0]
                  spart = gen.as_independent(symbol)[1].as_base_exp()[0]
                  if spart == symbol:
                      try:
                          poly = Poly(f_num, spart)
                      except PolynomialError:
                          pass
  
          result = False  # no solution was obtained
          msg = ""  # there is no failure message
  
          # Poly is generally robust enough to convert anything to
          # a polynomial and tell us the different generators that it
          # contains, so we will inspect the generators identified by
          # polys to figure out what to do.
  
          # try to identify a single generator that will allow us to solve this
          # as a polynomial, followed (perhaps) by a change of variables if the
          # generator is not a symbol
  
          try:
              if poly is None:
                  poly = Poly(f_num)
              if poly is None:
                  raise ValueError(
                      "could not convert %s to Poly" % f_num
                  )
          except GeneratorsNeeded:
              simplified_f = simplify(f_num)
              if simplified_f != f_num:
                  return _solve(simplified_f, symbol, **flags)
              raise ValueError("expression appears to be a constant")
  
          gens = [g for g in poly.gens if g.has(symbol)]
  
          def _as_base_q(x):
              """Return (b**e, q) for x = b**(p*e/q) where p/q is the leading
              Rational of the exponent of x, e.g. exp(-2*x/3) -> (exp(x), 3)
              """
              b, e = x.as_base_exp()
              if e.is_Rational:
                  return b, e.q
              if not e.is_Mul:
                  return x, 1
              c, ee = e.as_coeff_Mul()
              if c.is_Rational and c is not S.One:  # c could be a Float
                  return b**ee, c.q
              return x, 1
  
          if len(gens) > 1:
              # If there is more than one generator, it could be that the
              # generators have the same base but different powers, e.g.
              #   >>> Poly(exp(x) + 1/exp(x))
              #   Poly(exp(-x) + exp(x), exp(-x), exp(x), domain='ZZ')
              #
              # If unrad was not disabled then there should be no rational
              # exponents appearing as in
              #   >>> Poly(sqrt(x) + sqrt(sqrt(x)))
              #   Poly(sqrt(x) + x**(1/4), sqrt(x), x**(1/4), domain='ZZ')
  
              bases, qs = list(zip(*[_as_base_q(g) for g in gens]))
              bases = set(bases)
  
              if len(bases) > 1 or not all(q == 1 for q in qs):
                  funcs = {b for b in bases if b.is_Function}
  
                  trig = {
                      _
                      for _ in funcs
                      if isinstance(_, TrigonometricFunction)
                  }
                  other = funcs - trig
                  if not other and len(funcs.intersection(trig)) > 1:
                      newf = None
                      if f_num.is_Add and len(f_num.args) == 2:
                          # check for sin(x)**p = cos(x)**p
                          _args = f_num.args
                          t = a, b = [
                              i.atoms(Function).intersection(trig)
                              for i in _args
                          ]
                          if all(len(i) == 1 for i in t):
                              a, b = [i.pop() for i in t]
                              if isinstance(a, cos):
                                  a, b = b, a
                                  _args = _args[::-1]
                              if (
                                  isinstance(a, sin)
                                  and isinstance(b, cos)
                                  and a.args[0] == b.args[0]
                              ):
                                  # sin(x) + cos(x) = 0 -> tan(x) + 1 = 0
                                  newf, _d = (
                                      TR2i(_args[0] / _args[1]) + 1
                                  ).as_numer_denom()
                                  if not _d.is_Number:
                                      newf = None
                      if newf is None:
                          newf = TR1(f_num).rewrite(tan)
                      if newf != f_num:
                          # don't check the rewritten form --check
                          # solutions in the un-rewritten form below
                          flags["check"] = False
                          result = _solve(newf, symbol, **flags)
                          flags["check"] = check
  
                  # just a simple case - see if replacement of single function
                  # clears all symbol-dependent functions, e.g.
                  # log(x) - log(log(x) - 1) - 3 can be solved even though it has
                  # two generators.
  
                  if result is False and funcs:
                      funcs = list(
                          ordered(funcs)
                      )  # put shallowest function first
                      f1 = funcs[0]
                      t = Dummy("t")
                      # perform the substitution
                      ftry = f_num.subs(f1, t)
  
                      # if no Functions left, we can proceed with usual solve
                      if not ftry.has(symbol):
                          cv_sols = _solve(ftry, t, **flags)
                          cv_inv = list(
                              ordered(_vsolve(t - f1, symbol, **flags))
                          )[0]
                          result = [
                              {symbol: cv_inv.subs(sol)}
                              for sol in cv_sols
                          ]
  
                  if result is False:
                      msg = "multiple generators %s" % gens
  
              else:
                  # e.g. case where gens are exp(x), exp(-x)
                  u = bases.pop()
                  t = Dummy("t")
                  inv = _vsolve(u - t, symbol, **flags)
                  if isinstance(u, (Pow, exp)):
                      # this will be resolved by factor in _tsolve but we might
                      # as well try a simple expansion here to get things in
                      # order so something like the following will work now without
                      # having to factor:
                      #
                      # >>> eq = (exp(I*(-x-2))+exp(I*(x+2)))
                      # >>> eq.subs(exp(x),y)  # fails
                      # exp(I*(-x - 2)) + exp(I*(x + 2))
                      # >>> eq.expand().subs(exp(x),y)  # works
                      # y**I*exp(2*I) + y**(-I)*exp(-2*I)
                      def _expand(p):
                          b, e = p.as_base_exp()
                          e = expand_mul(e)
                          return expand_power_exp(b**e)
  
                      ftry = f_num.replace(
                          lambda w: w.is_Pow or isinstance(w, exp),
                          _expand,
                      ).subs(u, t)
                      if not ftry.has(symbol):
                          soln = _solve(ftry, t, **flags)
                          result = [
                              {symbol: i.subs(s)}
                              for i in inv
                              for s in soln
                          ]
  
          elif len(gens) == 1:
              # There is only one generator that we are interested in, but
              # there may have been more than one generator identified by
              # polys (e.g. for symbols other than the one we are interested
              # in) so recast the poly in terms of our generator of interest.
              # Also use composite=True with f_num since Poly won't update
              # poly as documented in issue 8810.
  
              poly = Poly(f_num, gens[0], composite=True)
  
              # if we aren't on the tsolve-pass, use roots
              if not flags.pop("tsolve", False):
                  soln = None
                  deg = poly.degree()
                  flags["tsolve"] = True
                  hints = ("cubics", "quartics", "quintics")
                  solvers = {h: flags.get(h) for h in hints}
                  soln = roots(poly, **solvers)
                  if sum(soln.values()) < deg:
                      # e.g. roots(32*x**5 + 400*x**4 + 2032*x**3 +
                      #            5000*x**2 + 6250*x + 3189) -> {}
                      # so all_roots is used and RootOf instances are
                      # returned *unless* the system is multivariate
                      # or high-order EX domain.
                      try:
                          soln = poly.all_roots()
                      except NotImplementedError:
                          if not flags.get("incomplete", True):
                              raise NotImplementedError(
                                  filldedent(
                                      """
      Neither high-order multivariate polynomials
      nor sorting of EX-domain polynomials is supported.
      If you want to see any results, pass keyword incomplete=True to
      solve; to see numerical values of roots
      for univariate expressions, use nroots.
      """
                                  )
                              )
                          else:
                              pass
                  else:
                      soln = list(soln.keys())
  
                  if soln is not None:
                      u = poly.gen
                      if u != symbol:
                          try:
                              t = Dummy("t")
                              inv = _vsolve(u - t, symbol, **flags)
                              soln = {
                                  i.subs(t, s)
                                  for i in inv
                                  for s in soln
                              }
                          except NotImplementedError:
                              # perhaps _tsolve can handle f_num
                              soln = None
                      else:
                          check = False  # only dens need to be checked
                      if soln is not None:
                          if len(soln) > 2:
                              # if the flag wasn't set then unset it since high-order
                              # results are quite long. Perhaps one could base this
                              # decision on a certain critical length of the
                              # roots. In addition, wester test M2 has an expression
                              # whose roots can be shown to be real with the
                              # unsimplified form of the solution whereas only one of
                              # the simplified forms appears to be real.
                              flags["simplify"] = flags.get(
                                  "simplify", False
                              )
                  if soln is not None:
                      result = [{symbol: v} for v in soln]
  
      # fallback if above fails
      # -----------------------
      if result is False:
          # try unrad
          if flags.pop("_unrad", True):
              try:
                  u = unrad(f_num, symbol)
              except (ValueError, NotImplementedError):
                  u = False
              if u:
                  eq, cov = u
                  if cov:
                      isym, ieq = cov
                      inv = _vsolve(ieq, symbol, **flags)[0]
                      rv = {
                          inv.subs(xi)
                          for xi in _solve(eq, isym, **flags)
                      }
                  else:
                      try:
                          rv = set(_vsolve(eq, symbol, **flags))
                      except NotImplementedError:
                          rv = None
                  if rv is not None:
                      result = [{symbol: v} for v in rv]
                      # if the flag wasn't set then unset it since unrad results
                      # can be quite long or of very high order
                      flags["simplify"] = flags.get("simplify", False)
              else:
                  pass  # for coverage
  
      # try _tsolve
      if result is False:
          flags.pop(
              "tsolve", None
          )  # allow tsolve to be used on next pass
          try:
              soln = _tsolve(f_num, symbol, **flags)
              if soln is not None:
                  result = [{symbol: v} for v in soln]
          except PolynomialError:
              pass
      # ----------- end of fallback ----------------------------
  
      if result is False:
          raise NotImplementedError("\n".join([msg, not_impl_msg % f]))
  
      result = _remove_duplicate_solutions(result)
  
      if flags.get("simplify", True):
          result = [{k: d[k].simplify() for k in d} for d in result]
          # Simplification might reveal more duplicates
          result = _remove_duplicate_solutions(result)
          # we just simplified the solution so we now set the flag to
          # False so the simplification doesn't happen again in checksol()
          flags["simplify"] = False
  
      if checkdens:
          # reject any result that makes any denom. affirmatively 0;
          # if in doubt, keep it
          dens = _simple_dens(f, symbols)
          result = [
              r
              for r in result
              if not any(checksol(d, r, **flags) for d in dens)
          ]
      if check:
          # keep only results if the check is not False
          result = [
              r
              for r in result
              if checksol(f_num, r, **flags) is not False
          ]
      return result
  
imports: from __future__ import annotations
  from sympy.core import S, Add, Symbol, Dummy, Expr, Mul
  from sympy.core.assumptions import check_assumptions
  from sympy.core.exprtools import factor_terms
  from sympy.core.function import expand_mul, expand_log, Derivative, AppliedUndef, UndefinedFunction, nfloat, Function, expand_power_exp, _mexpand, expand, expand_func
  from sympy.core.logic import fuzzy_not
  from sympy.core.numbers import Float, Rational, _illegal
  from sympy.core.intfunc import integer_log, ilcm
  from sympy.core.power import Pow
  from sympy.core.relational import Eq, Ne
  from sympy.core.sorting import ordered, default_sort_key
  from sympy.core.sympify import sympify, _sympify
  from sympy.core.traversal import preorder_traversal
  from sympy.logic.boolalg import And, BooleanAtom
  from sympy.functions import log, exp, LambertW, cos, sin, tan, acos, asin, atan, Abs, re, im, arg, sqrt, atan2
  from sympy.functions.combinatorial.factorials import binomial
  from sympy.functions.elementary.hyperbolic import HyperbolicFunction
  from sympy.functions.elementary.piecewise import piecewise_fold, Piecewise
  from sympy.functions.elementary.trigonometric import TrigonometricFunction
  from sympy.integrals.integrals import Integral
  from sympy.ntheory.factor_ import divisors
  from sympy.simplify import simplify, collect, powsimp, posify, powdenest, nsimplify, denom, logcombine, sqrtdenest, fraction, separatevars
  from sympy.simplify.sqrtdenest import sqrt_depth
  from sympy.simplify.fu import TR1, TR2i, TR10, TR11
  from sympy.strategies.rl import rebuild
  from sympy.matrices.exceptions import NonInvertibleMatrixError
  from sympy.matrices import Matrix, zeros
  from sympy.polys import roots, cancel, factor, Poly
  from sympy.polys.solvers import sympy_eqs_to_ring, solve_lin_sys
  from sympy.polys.polyerrors import GeneratorsNeeded, PolynomialError
  from sympy.polys.polytools import gcd
  from sympy.utilities.lambdify import lambdify
  from sympy.utilities.misc import filldedent, debugf
  from sympy.utilities.iterables import connected_components, generate_bell, uniq, iterable, is_sequence, subsets, flatten, sift
  from sympy.utilities.decorator import conserve_mpmath_dps
  from mpmath import findroot
  from sympy.solvers.polysys import solve_poly_system
  from types import GeneratorType
  from collections import defaultdict
  from itertools import combinations, product
  import warnings
  from sympy.solvers.bivariate import bivariate_type, _solve_lambert, _filtered_gens
label_desc: Use image node_id 8 for calling a global function with example usage: _solve(f) and returns: result, result, result, unknown, list, x,  int, list, b,  e, x,  int, unknown,  c, _solve, expand_power_exp