Solve three scheduling and array problems

In this round you are given three separate coding problems to solve. --- ### Problem 1: Schedule tasks with cooldown You are given: - A list of tasks, where each task is represented by a capital letter `'A'` to `'Z'`. - An integer `n >= 0`, the **cooldown period**. A single CPU can execute at most one task per time unit. The same type of task must be separated by at least `n` **idle intervals** (or other tasks) between two executions. You may insert **idle slots** (no task executed) if needed. Return the **minimum number of time units** required to execute all tasks. Example: - Input: `tasks = ['A','A','A','B','B','B']`, `n = 2` - One optimal schedule: `A B idle A B idle A B` - Output: `8` **Constraints**: - `1 <= tasks.length <= 10^4` - `0 <= n <= 10^4` Describe an efficient algorithm and then implement it. --- ### Problem 2: Remove duplicates from sorted array in-place You are given a sorted integer array `nums` (non-decreasing order). Remove the duplicates **in-place** such that each unique element appears only **once** and the relative order of elements is preserved. The function should return the **new length** `k` of the array containing only the unique elements. The first `k` elements of `nums` should hold the unique values; the remaining elements beyond `k` can be any value and are not important. You must use **O(1)** extra space. Example: - Input: `nums = [0,0,1,1,1,2,2,3,3,4]` - Output: - Return value: `5` - `nums` is modified to start with `[0,1,2,3,4, ...]` **Constraints**: - `0 <= nums.length <= 10^5` - `-10^9 <= nums[i] <= 10^9` Describe your approach (pointer technique, etc.) and implement the function. --- ### Problem 3: Rearrange string to avoid adjacent duplicates You are given a string `s` consisting of lowercase English letters. Rearrange the characters of `s` so that **no two adjacent characters are the same**. If it is not possible, indicate failure (for example, return an empty string or a special value as defined by the language/boilerplate). Example 1: - Input: `s = "aab"` - One valid output: `"aba"` Example 2: - Input: `s = "aaab"` - Output: impossible (because any arrangement has at least two adjacent `'a'`s). **Constraints**: - `1 <= s.length <= 10^5` - `s` contains only lowercase English letters `'a'` to `'z'`. Design an efficient algorithm (both time and space) and implement it.