
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
In a bamboo pole consisting of $n$ sections, the length of each section from top to bottom forms an arithmetic sequence. The length of the top section is 10 cm, and the total length of the last three sections is 114 cm. The length of the 6th section is the geometric mean of the first and the last sections. Find the value of $n$.

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