Simulate 2048 and pack board into uint64

Problem: 2048 move simulation + board compression

You are implementing part of the 2048 game on a fixed 4×4 grid.

Part A — Simulate a move

Given a 4×4 board of non-negative integers (each is either 0 or a power of two), implement a function:

• move(board, dir) -> new_board

where dir ∈ {LEFT, RIGHT, UP, DOWN}.

Rules (standard 2048):

1. All tiles slide as far as possible in the chosen direction.
2. If two adjacent tiles in the direction of movement have the same value, they merge into one tile of value 2x.
3. Each tile can merge at most once per move.
4. Zeros represent empty cells.

Part B — Encode/decode the board into a 64-bit integer

Implement two functions:

• encode(board) -> uint64
• decode(x) -> board

Constraints/assumptions for encoding:

• The board is 4×4 = 16 cells.
• Store each cell in 4 bits as the exponent e where value = 2^e.
• Empty cell (0) is stored as e = 0.
• Example: value 2 → e=1, 4 → e=2, …
• Assume no cell exceeds 2^15 (so exponent fits in 4 bits).
• You may choose any fixed cell order (e.g., row-major), but decode(encode(board)) must reproduce the original board.

Deliverables

Describe/implement move, encode, and decode with correct handling of merges and bit packing.