Circular Array Loop

You are playing a game involving a circular array of non-zero integers nums. Each nums[i] denotes the number of indices forward/backward you must move if you are located at index i:

• If nums[i] is positive, move nums[i] steps forward
• If nums[i] is negative, move |nums[i]| steps backward

Since the array is circular, you may assume that moving forward from the last element puts you on the first element, and moving backward from the first element puts you on the last element.

A cycle in the array consists of a sequence of indices seq of length k where:

• Following the movement rules above results in a repeating index sequence: seq[0] → seq[1] → ... → seq[k-1] → seq[0] → ...
• Every nums[seq[j]] is either all positive or all negative (the cycle must move in one consistent direction)
• k > 1 (the cycle length must be greater than 1)

Return true if there is a cycle in nums, or false otherwise.

Examples

Example 1:

Input: nums = [2,-1,1,2,2]

Output: true

Explanation: There is a cycle: index 0 → 2 → 3 → 0. The cycle length is 3 and all values at these indices (2, 1, 2) are positive.

Example 2:

Input: nums = [-1,2]

Output: false

Explanation: The sequence index 1 → 1 → 1 → ... is a cycle of length 1, so it does not count. The sequence index 0 → index 1 → index 0 has mixed directions (negative then positive), so it does not count.

Example 3:

Input: nums = [-2,1,-1,-2,-2]

Output: false

Explanation: The sequence index 1 → 2 → 1 has mixed signs (positive then negative). The sequence index 3 → 1 also has mixed signs.

Constraints

• 1 <= nums.length <= 5000
• -1000 <= nums[i] <= 1000
• nums[i] != 0