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Our goal is to return `True` if `nums` contains a "good subarray" or `False` otherwise.

### Q: What is the concept of a "good subarray"? Could you explain it in more detail?

A "good subarray" is defined as follows:

1. It is a subarray of `nums` with a length of at least 2.
2. The sum of the elements in this subarray is a multiple of `k` (meaning the sum divided by `k` leaves a remainder of 0).

This is actually a common type of problem called the **"Continuous Subarray Sum"** problem. Here, our task is to check if there exists a subarray within `nums` that meets these criteria.

---

**Question: Some notes:**

1. A subarray is a contiguous part of an array — meaning the elements are consecutive.
2. An integer is a multiple of \( k \) if there exists an integer \( n \) such that:
    
    \[
    x = n \times k
    \]
    

**Question: What is the input and output format?**

- **Input**: `List[int]`, `k: int`
- **Output**: `bool`

**Question: Can you walk us through an example?**

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**Example:**

We have an example with the array `nums = [23, 2, 4, 6, 7]` and `k = 6`.

First, we’ll start with a naive solution and then see how we can optimize it.

Our goal is to generate all subarrays of `nums` with a length of 2 or more.

Sure, here are all the subarrays of `nums = [23, 2, 4, 6, 7]` with length ≥ 2:

1. **Starting from index 0:**
    - `[23, 2]`
    - `[23, 2, 4]`
    - `[23, 2, 4, 6]`
    - `[23, 2, 4, 6, 7]`
2. **Starting from index 1:**
    - `[2, 4]`
    - `[2, 4, 6]`
    - `[2, 4, 6, 7]`
3. **Starting from index 2:**
    - `[4, 6]`
    - `[4, 6, 7]`
4. **Starting from index 3:**
    - `[6, 7]`

These are all the possible subarrays of length ≥ 2 for `nums = [23, 2, 4, 6, 7]`.

Let me know if you'd like to proceed with further steps or optimizations!

Here's the text formatted and translated into clear English:

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**Question:** Let's continue with the naive solution for now.

**Question:** What do we need to do to compute the cumulative sum of each subarray here?

**Answer:** I think the most optimized approach is to use a separate array called `prefix_sum` to store the cumulative sum up to each index \( i \).

For example, if `nums = [23, 2, 4, 6, 7]`, the cumulative sum array (prefix sum) would be `[23, 25, 29, 35, 42]`.

**Question:** Next, after generating all the possible valid subarrays...

**Question:** And also after generating the prefix sum, will it help us efficiently compute the sum of elements within the boundaries of the subarrays?

**Answer:** Yes, but we have one trick left!

We need to keep track of the starting position of each subarray. So we need a data structure or process to help us track the starting position of each subarray.

**Question:** But why do we need it?

**Answer:** It's important because when identifying a valid subarray, we need to use the prefix sum array to compute the sum. For this, we need to know the left (lower bound) and right (upper bound) indices of the subarray.

By knowing just the starting position (left bound), we can generate the right boundaries efficiently.

Q- so far i use a lot of abract lanabuage oki 

Q- i translate my thinking porcess so far so can you give a good feadback about it where i done it right and where not ?