SMO 2026 Senior Round 1 Question 4

Let $\{a_n\}_{n=1}^{m}$ be a finite sequence of real numbers. Find the largest value of $m$ such that $\sum_{k=0}^{4}a_{i+k}\lt 0$ and $\sum_{k=0}^{7}a_{j+k}>0$ for any positive integers $i,j$.