
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
The sequence \( \{a_{n}\} \) satisfies \( a_{1} = 1 \), \( a_{2} = 2 \), \( a_{3} = 3 \), \( a_{4} = 4 \), \( a_{5} = 5 \). For \( n \geq 5 \), \( a_{n+1} = a_{1} a_{2} \cdots a_{n} - 1 \). How many positive integers \( m \) exist such that \( a_{1}a_{2}\cdots a_{m} = a_{1}^{2} + a_{2}^{2} + \cdots + a_{m}^{2} \)?

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