What changes at every step
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Step 1PATH -> PATE
Tile 4 changes H to E; the move is sideways by letter shape. PATE has 16 legal neighbors, 2 shortest continuations, and 5 moves remaining. Other legal exits from PATH include BATH, PATS, MATH, OATH.
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Step 2PATE -> PALE
Tile 3 changes T to L; the move is sideways by letter shape. PALE has 22 legal neighbors, 2 shortest continuations, and 4 moves remaining. Other legal exits from PATE include PARE, PAGE, DATE, PACE.
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Step 3PALE -> POLE
Tile 2 changes A to O; the move is closer by letter shape. POLE has 18 legal neighbors, 2 shortest continuations, and 3 moves remaining. Other legal exits from PALE include PALL, PALM, PILE, PALS.
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Step 4POLE -> POLL
Tile 4 changes E to L; the move is sideways by letter shape. POLL has 12 legal neighbors, 1 shortest continuation, and 2 moves remaining. Other legal exits from POLE include POME, POPE, POLO, POLY.
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Step 5POLL -> 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, PILL.
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Step 6POOL -> 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.