Brownian Martingale II

Let $W_t$ be a standard Brownian Motion. Define $X_t = W_t^n - \int_0^t g(W_s)~ds$ for some continuous-time stochastic process $g(W_s)$ that is adapted to the Brownian Motion filtration. Find the process $g(W_s)$ that makes $X_t$ a martingale. Report the value of $\mathbb{E}\left[ g(W_t) \right]$ when $n = 6$ and $t = 1$.