Question 4
Fix a positive integer $N$. Let $a_k,b_k$ $(0\le k\le N)$ and $c_k$ $(1\le k\le N)$ be sequences of real numbers such that for all $1\le k\le N$, we have
$$a_k=\frac{4a_{k-1}+3b_{k-1}}5+c_k,$$
$$b_k=\frac{4b_{k-1}-3a_{k-1}}5.$$
If $a_0=b_0=0$ and $a_N\ge1$, determine the smallest possible value of
$$|c_1|+|c_2|+\cdots+|c_N|.$$
$$a_k=\frac{4a_{k-1}+3b_{k-1}}5+c_k,$$
$$b_k=\frac{4b_{k-1}-3a_{k-1}}5.$$
If $a_0=b_0=0$ and $a_N\ge1$, determine the smallest possible value of
$$|c_1|+|c_2|+\cdots+|c_N|.$$