For $X_i\sim N(\mu,\sigma^2)$, how to find the MSE of estimator: $\hat \sigma^2 = \frac 1{n+2}\sum_{i=1}^nX_i^2$.

Question

This is part of a larger problem. I have already found the MSE of two other estimators: $\hat\sigma_1^2 = \frac1{n-1}\sum_i^n (X_i-\bar X)^2$ and $\hat\sigma_2^2 = \frac1{n+1}\sum_i^n (X_i-\bar X)^2$. Now I need to find the MSE of $\frac1{n+2}\sum_{i=1}^n X_i^2$.

I have tried breaking the problem down into $\text{MSE} = \text{Var} + \text{Bias}^2$, but I have been unable to find the variance. I am starting to think that I am not approaching this problem in the best way. Maybe I should somehow exploit the earlier results? Any tips would be appreciated.