
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
Consider the integer sequence \( a_{1}, a_{2}, \cdots, a_{10} \) satisfying:
\[
a_{10} = 3a_{1}, \quad a_{2} + a_{8} = 2a_{5}
\]
and \( a_{i+1} \in \{1 + a_{i}, 2 + a_{i}\} \) for \(i = 1, 2, \cdots, 9\). Find the number of such sequences.

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