Check if array is sorted and rotated leetcode 1752

okay, let's break down leetcode problem 1752: "check if array is sorted and rotated." i'll provide a comprehensive tutorial with explanations, strategies, and code examples in multiple languages (python, java, and c++).

**understanding the problem**

the problem states that you are given an array `nums`. you need to determine if `nums` can be obtained by rotating a sorted array. in other words, you're looking to see if the array is "almost sorted" in a way where there's a single point of discontinuity.

**formal definition:**

an array `nums` is considered sorted and rotated if:

1. it's possible to sort `nums` in non-decreasing order.
2. there exists an index `k` (0 = k `n`, where `n` is the length of `nums`) such that rotating the sorted array `k` positions results in `nums`.

**example cases:**

* **example 1:** `nums = [3, 4, 5, 1, 2]`
* the sorted version is `[1, 2, 3, 4, 5]`.
* rotating the sorted array by 3 positions gives us `[3, 4, 5, 1, 2]`.
* **output:** `true`

* **example 2:** `nums = [2, 1, 3, 4]`
* no matter how you rotate a sorted version of `[1, 2, 3, 4]`, you cannot obtain `[2, 1, 3, 4]`.
* **output:** `false`

* **example 3:** `nums = [1, 2, 3]`
* the array is already sorted. a rotation of 0 positions results in the same array.
* **output:** `true`

* **example 4:** `nums = [7,9,11,13,3,5]`
* the sorted array would be [3,5,7,9,11,13]
* rotating by 4 produces the array.
* **output:** true

**core idea and algorithm**

the key insight is that in a sorted and rotated array, there will be **at most one** point where the order is broken. that is, at most one instance where `nums[i] nums[i+1]`.

here's the step-by-step algorithm:

1. **count the inversions:** iterate through the array and count the number of times `nums[i] nums[i+1]`. make sure to handle the circular case by comparing the last element with the first element (`nums[n-1] nums[0]`). this handles the "wrap-ar ...

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