
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  $20^{21} = 2^a5^b = 4^c5^d = 8^e5^f$  for positive integers  $a,b,c,d,e,$  and  $f$ . Find  $\frac{100bdf}{ace}$ .

*Proposed by Andrew Wu*

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