Does the existence of solutions of $P(x)\equiv 0\pmod{\mathfrak{m}^n}$ for all $n$ gives a solution of $P(x)=0$?

Question

Let $A$ be a complete local noetherian ring with maximal ideal $\mathfrak{m}$. Let $P(X_1,...,X_d)$ be a polynomial in $d$ variables and coefficients in $A$ and assume that the equation $$P(x_1,...,x_d)\equiv 0\pmod{\mathfrak{m}^n}$$ admits a solution for all $n\geq 1$. Can I conclude that the equation $P(x_1,...,x_d)=0$ admits a solution in $A^d$?

The problem lies in the fact that it is not clear how we can find a compatible system of solutions.

Many thanks!