Binomial Theorem, Pascal & Combinatorial Identities · Superchallenge problem

The Fibonacci numbers are $F_0=0$, $F_1=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for $n\ge2$. Evaluate $\displaystyle\sum_{k=0}^{12}\binom{12}{k}F_k$.