
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
On a $n \times m$ chessboard, a piece moves from the bottom left corner to either the square just above, the square just right, or the diagonal square just right-top. In how many cases can the first player guarantee to win, given that the chessboards are $6\times 7$, $6\times 8$, $7\times 7$, $7\times 8$, and $8\times 8$?

Present the answer in LaTex format: \boxed{Your answer}
user