
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
The sequence \(a_n\) is constructed as follows: \(a_1, a_2\) are arbitrary real numbers; for \(n \geq 3\), the number \(a_n\) is the smallest of the numbers \(\left|a_i-a_j\right|\) for \(1 \leq i < j \leq n-1\). For example, if \(a_1=6\) and \(a_2=\frac{19}{2}\), then the sequence becomes \(6, \frac{19}{2}, \frac{7}{2}, \frac{5}{2}, 1, 1, 0, 0, 0, \ldots\). For some choice of \(a_1\) and \(a_2\), a sequence is obtained in which \(a_{10}=1\). Find the smallest possible value of \(a_3\) in such a sequence.

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