
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
Will has a magic coin that can remember previous flips. If the coin has already turned up heads  $m$  times and tails  $n$  times, the probability that the next flip turns up heads is exactly  $\frac{m+1}{m+n+2}$ . Suppose that the coin starts at  $0$  flips. The probability that after  $10$  coin flips, heads and tails have both turned up exactly  $5$  times can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $m+n$ .

*Proposed by Nathan Xiong*

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