You are given an integer array nums of length $3$, where each element represents the length of a side. Determine what kind of triangle these three sides form and return the corresponding string.

Return:

• "equilateral" if all three sides are equal

• "isosceles" if exactly two sides are equal

• "scalene" if all three sides are different

• "none" if the three sides cannot form a valid triangle

A set of side lengths forms a valid triangle if the sum of any two sides is strictly greater than the remaining side.

Note: A triangle is valid only when all three pairwise triangle inequalities hold.

Constraints:

• nums.length == 3

• $1 \leq$ nums[i] $\leq 100$

Input: nums = [3, 3, 3]-List[int]
Output: "equilateral"-str
Explanation: All three sides are equal, and the triangle inequalities hold (e.g., 3 + 3 > 3).

Input: nums = [4, 4, 6]-List[int]
Output: "isosceles"-str
Explanation: Exactly two sides are equal (4 and 4), and the triangle is valid since 4 + 4 > 6, 4 + 6 > 4, and 4 + 6 > 4.

Input: nums = [4, 5, 6]-List[int]
Output: "scalene"-str
Explanation: All three sides are different, and the triangle is valid because each pair sums to more than the remaining side.

Input: nums = [1, 2, 3]-List[int]
Output: "none"-str
Explanation: This is not a valid triangle because 1 + 2 is not strictly greater than 3 (it equals 3), violating the strict triangle inequality.

Input: nums = [2, 3, 10]-List[int]
Output: "none"-str
Explanation: This is not a valid triangle because 2 + 3 ≤ 10, so the sides cannot form a triangle.

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