Maximize protected population and bitwise AND

You are given two independent coding problems.

Problem 1: Maximize protected population after moving units

There are n cities in a line (indexed 1..n). City i has population population[i].

Initially, some cities contain one security unit and others contain none. You are given an array unit[i] where:

• unit[i] = 1 means city i initially has one unit
• unit[i] = 0 means it has no unit

You may reassign units using these rules:

• Each security unit may move at most one city to the left (from city i to city i-1 ).
• Each security unit can be moved at most once (so it either stays, or moves left by 1).
• A unit in city 1 cannot move left.
• After all moves, a city is considered protected if it contains at least one unit (multiple units in one city are allowed but do not add extra benefit).

Goal: After performing moves, maximize the total population of protected cities.

Task: Return the maximum possible protected population sum.

Problem 2: Maximize bitwise AND after at most k increments

You are given an array customer_rating of length n (assume all values are non-negative integers).

You may perform the following operation at most k times in total:

• Choose any index i and do customer_rating[i] += 1 .

After performing up to k increments, you must:

• Choose exactly m elements from the final array (any indices you want), and
• Compute the bitwise AND of those m chosen values. Call the result new_rating .

Goal: Maximize new_rating.

Task: Return the maximum possible new_rating achievable with at most k total +1 operations.