
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 positive numbers \( a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \) satisfy the following: \( a_{1}, a_{2}, a_{3} \) form an arithmetic sequence with common difference \( b_{1} \); \( b_{1}, b_{2}, b_{3} \) form a geometric sequence with common ratio \( a_{1} \); and \( a_{3} = b_{3} \). Find the minimum value of \( a_{3} \) and determine the value of \( a_{2} b_{2} \) when \( a_{3} \) reaches its minimum value.

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