
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, and for \(n \geq 3\), the number \(a_{n}\) is the smallest of the numbers \(\left|a_{i}-a_{j}\right|\), \(1 \leq i < j \leq n-1\). For example, if \(a_{1}=6, a_{2}=\frac{19}{2}\), we get the sequence \(6, \frac{19}{2}, \frac{7}{2}, \frac{5}{2}, 1, 1, 0, 0, 0, \ldots\). For some choice of \(a_{1}, a_{2}\), the sequence obtained has \(a_{10}=1\). Find the smallest possible value of \(a_{3}\) in such a sequence.

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