
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
Suppose that  $a_1, a_2, a_3, \ldots$  is an infinite geometric sequence such that for all  $i \ge 1$ ,  $a_i$  is a positive integer. Suppose furthermore that  $a_{20} + a_{21} = 20^{21}$ . If the minimum possible value of  $a_1$  can be expressed as  $2^a 5^b$  for positive integers  $a$  and  $b$ , find  $a + b$ .

*Proposed by Andrew Wu*

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