Inverse drift SDE solution

Question

Does the following SDE have a solution:

$$ dX_t = \frac{dt}{X_t}+W_t $$

where $W_t$ is a Brownian Motion, with initial state $X_0$?

I have tried applying Ito's lemma to $X_t^2$:

$$ dX_t^2 = 3dt + 2X_tdW_t $$

But I am again blocked.