
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 \( a_{1}, a_{2}, \ldots, a_{11} \) be positive integers with \( a_{1} < a_{2} < \cdots < a_{11} \). If \( \frac{1}{a_{1}}, \frac{1}{a_{2}}, \ldots, \frac{1}{a_{11}} \) form an arithmetic sequence (i.e., \( \frac{1}{a_{2}} - \frac{1}{a_{1}} = \frac{1}{a_{3}} - \frac{1}{a_{2}} = \cdots = \frac{1}{a_{11}} - \frac{1}{a_{10}} \) ), find the smallest possible value of \( a_{1} \).

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