
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 $T$ be the set of all ordered triples of integers $(b_1, b_2, b_3)$ with $1 \leq b_1, b_2, b_3 \leq 20$. Each ordered triple in $T$ generates a sequence according to the rule $b_n = b_{n-1} \cdot |b_{n-2} - b_{n-3}|$ for all $n \geq 4$. Find the number of such sequences for which $b_n = 0$ for some $n$.

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