
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
Given an integer sequence $a_{1}, a_{2}, \cdots, a_{10}$ satisfying $a_{10}=3 a_{1}$, $a_{2}+a_{8}=2 a_{5}$, and
$$
a_{i+1} \in\left\{1+a_{i}, 2+a_{i}\right\}, i=1,2, \cdots, 9,
$$
find the number of such sequences.

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