Defining the DP State

You have a directed graph with weighted edges (can be negative). Find the maximum-score path from $1$ to $n$. If there is a positive cycle reachable from $1$ that can reach $n$, the answer is infinite. Otherwise, use DP on the DAG formed by nodes reachable from $1$.

Process in topological order, track dp[v] = max score to reach $v$. Detect positive cycles using a modified DFS that tracks nodes in the current recursion stack.