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Okay, let's dive deep into the "Maximum Average Subarray I" problem, exploring various approaches, providing detailed explanations, and including code examples in multiple languages.

*Problem Statement*

You are given an array of integers `nums` and an integer `k`.

Find the contiguous subarray (containing at least `k` elements) which has the maximum average value. And you need to output the maximum average value.

*Example:*



*Understanding the Problem*

The core task is to find a subarray of length `k` within `nums` that yields the highest possible average. This involves calculating the average for every possible subarray of size `k` and tracking the maximum.

*Approaches*

1. *Brute Force (Naive Approach)*

*Idea:* Iterate through all possible starting positions of a subarray of length `k` and calculate the average for each. Keep track of the maximum average found so far.

*Algorithm:*
1. Initialize `max_avg` to negative infinity (or a very small number to ensure it gets updated).
2. Outer loop: Iterate from `i = 0` to `len(nums) - k + 1`. This is the starting index of the subarray.
3. Inner loop: Calculate the sum of the subarray `nums[i:i+k]`.
4. Calculate the average of the subarray: `sum / k`.
5. Update `max_avg` if the current average is greater.
6. Return `max_avg`.

*Code (Python):*



*Code (Java):*



*Time Complexity:* O(n*k) where n is the length of `nums`. The outer loop runs approximately `n` times, and the inner loop runs `k` times.
*Space Complexity:* O(1) Constant extra space.

*Drawbacks:* This is the most straightforward approach but has the worst time complexity. It's inefficient for large arrays and values of `k`. It will likely lead to a "Time Limit Exceeded" (TLE) error on many online coding platforms.

2. *Sliding Window (Optimized)*

*Idea:* Instead of recalculating th ...

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