
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 \( a_{1}, a_{2}, \ldots \) be an arithmetic sequence and \( b_{1}, b_{2}, \ldots \) be a geometric sequence. Suppose that \( a_{1} b_{1}=20 \), \( a_{2} b_{2}=19 \), and \( a_{3} b_{3}=14 \). Find the greatest possible value of \( a_{4} b_{4} \).

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