The function f sends sequences to sequences in the following way: given a sequence \{a_{n}\}_{n=0}^{\infty} of real numbers, f sends \{a_{n}\}_{n=0}^{\infty} to the sequence \{b_{n}\}_{n=0}^{\infty}, where b_{n}=\sum_{k=0}^{n} a_{k}\binom{n}{k} for all n \geq 0. Let \{F_{n}\}_{n=0}^{\infty} be the Fibonacci sequence, defined by F_{0}=0, F_{1}=1, and F_{n+2}=F_{n+1}+F_{n} for all n \geq 0. Let \{c_{n}\}_{n=0}^{\infty} denote the sequence obtained by applying the function f to the sequence \{F_{n}\}_{n=0}^{\infty} 2022 times. Find c_{5}(\bmod 1000).