
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 $1 \leq a_1, a_2, a_3 \leq 15$. Each ordered triple in $S$ generates a sequence according to the rule $a_n = 2 \cdot a_{n-1} \cdot |a_{n-2} - a_{n-3}|$ for all $n \geq 4$. Find the number of such sequences for which $a_n = 0$ for some $n$.

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