
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 sequence \(a_{1}, a_{2}, a_{3}, \cdots, a_{21}\) satisfy \(\left|a_{n+1}-a_{n}\right| = 1\) for \(n = 1, 2, 3, \cdots, 20\), and \(a_{1}, a_{7}, a_{21}\) form a geometric sequence. If \(a_{1}=1\) and \(a_{21}=9\), find the number of distinct sequences that satisfy the conditions.

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