
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
Given \( a_{1}, a_{2}, a_{3} \) form an arithmetic sequence with \( a_{1} + a_{2} + a_{3} = 15 \), and \( b_{1}, b_{2}, b_{3} \) form a geometric sequence with \( b_{1} b_{2} b_{3} = 27 \). If \( a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3} \) are positive integers and form a geometric sequence, find the maximum value of \( a_{3} \).

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