Paginate forward and backward through results

You are asked to extend a basic pagination system to support forward and backward navigation, similar to a cursor-based API.

You are given:

• A zero-indexed array of items items sorted in a fixed, deterministic order.
• An integer pageSize > 0.

Design a paginator with the following operations:

1. getFirstPage() → returns the first pageSize items and a cursor representing the position after the last returned item.
2. getNextPage(cursor) → given a cursor returned by a previous call, return the next pageSize items and a new cursor. If there are no more items, return an empty list and a special "end" cursor.
3. getPrevPage(cursor) → given a cursor, return the previous page of pageSize items and a new cursor pointing to the start of that page. If there is no previous page, return an empty list and a "start" cursor.

Requirements:

• Define what the cursor contains (it could just be an index or a more complex token).
• All operations should run in O(pageSize) time.
• Handle edge cases correctly, including:
  • Moving past the last page with getNextPage.
  • Moving before the first page with getPrevPage.
  • Cases where len(items) is not a multiple of pageSize.

Implement the data structure and its methods with clear handling of indices and cursors.

Cursor Pagination Forward and Backward

Implement solution(items, page_size, operations). A cursor is the integer index after the last item returned by a page. Operations are ["first"], ["next", cursor], and ["prev", cursor]. Return [page_items, new_cursor] for each operation.

Constraints

• page_size > 0
• items are already sorted in deterministic order

Examples

Example 1

Input: (['a', 'b', 'c', 'd', 'e'], 2, [['first'], ['next', 2], ['next', 4], ['next', 5], ['prev', 5], ['prev', 1], ['prev', 0]])
Output: [[['a', 'b'], 2], [['c', 'd'], 4], [['e'], 5], [[], 5], [['d', 'e'], 3], [['a'], 0], [[], 0]]

Example 2

Input: ([1, 2, 3], 5, [['first'], ['next', 3], ['prev', 3]])
Output: [[[1, 2, 3], 3], [[], 3], [[1, 2, 3], 0]]

Hints

1. Use the cursor as an array boundary.
2. Clamp invalid cursors into [0, len(items)].