
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  $S$  be the set of all ordered triples of integers  $(a_1,a_2,a_3)$  with  $1 \le a_1,a_2,a_3 \le 10$ . Each ordered triple in  $S$  generates a sequence according to the rule  $a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |$  for all  $n\ge 4$ . Find the number of such sequences for which  $a_n=0$  for some  $n$ .

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