Solve Prime Jumps and Pipeline Scaling

An online assessment contains two independent coding problems. Solve both.

Problem 1: Prime-Constrained Score Jump

You are given an integer array score of length n and an integer k. You start at index 0, and your initial total is score[0]. On each move, you may jump from index i to index j only if:

• j > i,
• j - i <= k, and
• j - i is a prime number.

When you land on an index, you add score[j] to your total. Return the maximum total score you can obtain when reaching index n - 1. If index n - 1 cannot be reached, return null.

Assume score may contain negative values.

Problem 2: Maximize Pipeline Throughput

You are given a serial processing pipeline with n services. Service i has:

• base throughput t[i], and
• expansion cost cost[i].

The pipeline throughput is the minimum throughput among all services.

You may expand any service any nonnegative integer number of times. If service i is expanded x times, then:

• its throughput becomes t[i] * (1 + x), and
• the total money spent on that service is x * cost[i].

Given a total budget budget, return the maximum possible overall pipeline throughput.