
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  $a_1, a_2, \dots$  be an arithmetic sequence and  $b_1, b_2, \dots$  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}
user