Problem Description

You are given an integer array a of length n. An operation is defined as follows:

1. Remove the first element, x = a[0].
2. Append x to the end of the array.
3. Starting from its new position at the end, repeatedly swap x leftward. This swap continues as long as x has a left neighbor and x is strictly smaller than that left neighbor (a[i] < a[i-1]). The process stops when either x reaches index 0 or x is no longer strictly smaller than its left neighbor (a[i-1] <= a[i]).

After each operation, the array is updated deterministically.

Task

Return the minimum number of operations required to make the array non-decreasing. An array is non-decreasing if, for all valid indices i, a[i] <= a[i+1].

If it is impossible for the array to ever become non-decreasing under repeated operations, return -1.

Function Signature

Implement a function with the following signature:

min_ops_to_nondecreasing(a: List[int]) -> int

Constraints

• 1 <= n <= 2e5
• a[i] fits within a 32-bit signed integer.
• The time complexity of the solution must be better than simulating all full array states.