
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
Don Miguel places a token in one of the  $(n+1)^2$  vertices determined by an  $n \times n$  board. A *move* consists of moving the token from the vertex on which it is placed to an adjacent vertex which is at most  $\sqrt2$  away, as long as it stays on the board. A *path* is a sequence of moves such that the token was in each one of the  $(n+1)^2$  vertices exactly once. What is the maximum number of diagonal moves (those of length  $\sqrt2$ ) that a path can have in total?

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