
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
Given a grid of  $6\times 6$ , write down integer which is more than or equal to 1 and less than or equal to 6 in each grid. For integers  $1\leq i,\ j\leq 6$ , denote
denote by  $i \diamondsuit j$  the integer written in the entry in the  $i-$  th row and  $j-$  th colum of the grid.

How many ways are there to write down integers satisfying the condition as follows ? $\bullet$  For any integer  $i$  with  $1\leq i\leq 6$ ,  $i \diamondsuit i=i$  holds. $\bullet$  For any integer  $i,\ j,\ k,\ l$  with  $1\leq i,\ j,\ k,\ l\leq 6,\ (i \diamondsuit j) \diamondsuit (k \diamondsuit l)=i \diamondsuit l$  holds.

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