
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
Let  $a_1,a_2,\cdots ,a_9$  be  $9$  positive integers (not necessarily distinct) satisfying: for all  $1\le i<j<k\le 9$ , there exists  $l (1\le l\le 9)$  distinct from  $i,j$  and  $j$  such that  $a_i+a_j+a_k+a_l=100$ . Find the number of  $9$ -tuples  $(a_1,a_2,\cdots ,a_9)$  satisfying the above conditions.

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