
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
A  $3 \times 3$  grid of unit cells is given. A *snake of length  $k$* is an animal which occupies an ordered  $k$ -tuple of cells in this grid, say  $(s_1, \dots, s_k)$ . These cells must be pairwise distinct, and  $s_i$  and  $s_{i+1}$  must share a side for  $i = 1, \dots, k-1$ . After being placed in a finite  $n \times n$  grid, if the snake is currently occupying  $(s_1, \dots, s_k)$  and  $s$  is an unoccupied cell sharing a side with  $s_1$ , the snake can *move* to occupy  $(s, s_1, \dots, s_{k-1})$  instead. The snake has *turned around* if it occupied  $(s_1, s_2, \dots, s_k)$  at the beginning, but after a finite number of moves occupies  $(s_k, s_{k-1}, \dots, s_1)$  instead.

Find the largest integer  $k$  such that one can place some snake of length  $k$  in a  $3 \times 3$  grid which can turn around.

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