The first thing I'd do is get the (well known) distributions of U1 = X1 + X2 + X3 and U2 = X4 + X5 + X6.

Then I'd think about the distributions of V1 = U1^2 and V2 = U2^2, and I'd also be thinking about the well known distribution of X^2 .....

Then I'd be thinking about the distribution of W = V1 + V2, in light of the above.