
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 the integer sequence \(a_1, a_2, \ldots, a_{10}\) satisfy \(a_{10} = 3a_1\) and \(a_2 + a_8 = 2a_5\), with \(a_{i+1} \in \{1 + a_i, 2 + a_i\}\) for \(i = 1, 2, \ldots, 9\). How many such sequences exist?

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