Maximum Score with Prime Jumps

Problem Description

Given an array arr of n integers, you are playing a game starting at index 0. You want to reach the last index n-1 to maximize the sum of values at the cells you land on (including the start and end).

You can move from your current index i to index j (where j > i) under the following conditions:

1. Step move: You can move to i + 1.
2. Prime jump: You can jump to i + k if k is a prime number and the jump lands on an index j where arr[j] ends with the digit 3 (i.e., arr[j] % 10 == 3).

Your goal is to find the maximum possible sum of values encountered along the path from index 0 to n-1. If it is impossible to reach the end, return -1.

Function Signature

def max_score(arr: List[int]) -> int:

Constraints

• 2 <= len(arr) <= 5000
• 1 <= arr[i] <= 10^5
• The prime jumps must be valid indices within bounds.

Examples

Example 1: Input: arr = [1, 5, 13, 2, 3] Output: 17 Explanation: Start at index 0 (val 1). Move to index 1 (val 5). Jump to index 2 (val 13, prime jump of 1 is not allowed, but index 2 is reachable via step/prime logic). A valid path is 0 -> 1 -> 2 -> 4. Sum: 1 + 5 + 13 + 3 = 22.

Example 2: Input: arr = [10, 2, 8] Output: -1 Explanation: No valid path exists to reach the end.

Edge Cases

• Path leads to a dead end.
• No primes exist that satisfy the jump condition.