
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 that $\{a\_n\}$ is an arithmetic sequence with a non-zero common difference, and $a\_9$, $a\_1$, $a\_5$ form a geometric sequence. Also, $a\_1 + 3a\_5 + a\_9 = 20$. Find the value of $a\_{13}$.

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