What changes at every step
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Step 1KILT -> KILL
Tile 4 changes T to L; the move is sideways by letter shape. KILL has 14 legal neighbors, 1 shortest continuation, and 4 moves remaining. Other legal exits from KILT include HILT, TILT, KILN, WILT.
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Step 2KILL -> PILL
Tile 1 changes K to P; the move is closer by letter shape. PILL has 15 legal neighbors, 1 shortest continuation, and 3 moves remaining. Other legal exits from KILL include WILL, BILL, HILL, FILL.
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Step 3PILL -> POLL
Tile 2 changes I to O; the move is closer by letter shape. POLL has 12 legal neighbors, 1 shortest continuation, and 2 moves remaining. Other legal exits from PILL include PULL, PALL, WILL, BILL.
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Step 4POLL -> POOL
Tile 3 changes L to O; the move is closer by letter shape. POOL has 8 legal neighbors, 1 shortest continuation, and 1 move remaining. Other legal exits from POLL include ROLL, TOLL, PULL, DOLL.
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Step 5POOL -> POOP
Tile 4 changes L to P; the move is closer by letter shape. POOP has 11 legal neighbors, target reached, and 0 moves remaining. Other legal exits from POOL include POOR, POOF, TOOL, COOL.