
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
A coin that comes up heads with probability  $ p > 0$  and tails with probability  $ 1\minus{}p > 0$  independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to  $ \frac{1}{25}$  of the probability of five heads and three tails. Let  $ p \equal{} \frac{m}{n}$ , where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m\plus{}n$ .

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