def METHOD_NAME(self, problem, dependencies):
    """Replaces every coefficient in the residual and jacobian with a deepcopy to return
    a clone of the original NonlinearVariationalProblem instance. We'll be modifying the
    numerical values of the coefficients in the residual and jacobian, so in order not to
    affect the user-defined self._ad_problem.F, self._ad_problem.J and self._ad_problem.u
    expressions, we'll instead create clones of them.
    """
    from firedrake import Function, NonlinearVariationalProblem
    F_replace_map = {}
    J_replace_map = {}
    F_coefficients = problem.F.coefficients()
    J_coefficients = problem.J.coefficients()
    _ad_count_map = {}
    for block_variable in dependencies:
        coeff = block_variable.output
        if coeff in F_coefficients and coeff not in F_replace_map:
            if isinstance(coeff, Function) and coeff.ufl_element().family() == "Real":
                F_replace_map[coeff] = copy.deepcopy(coeff)
            else:
                F_replace_map[coeff] = coeff.copy(deepcopy=True)
            _ad_count_map[F_replace_map[coeff]] = coeff.count()
        if coeff in J_coefficients and coeff not in J_replace_map:
            if coeff in F_replace_map:
                J_replace_map[coeff] = F_replace_map[coeff]
            elif isinstance(coeff, Function) and coeff.ufl_element().family() == "Real":
                J_replace_map[coeff] = copy.deepcopy(coeff)
            else:
                J_replace_map[coeff] = coeff.copy()
            _ad_count_map[J_replace_map[coeff]] = coeff.count()
    nlvp = NonlinearVariationalProblem(replace(problem.F, F_replace_map),
                                       F_replace_map[problem.u],
                                       bcs=problem.bcs,
                                       J=replace(problem.J, J_replace_map))
    nlvp._ad_count_map_update(_ad_count_map)
    return nlvp