
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
When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to  $0$  and is the same as that of getting heads exactly twice. Let  $\frac ij$ , in lowest terms, be the probability that the coin comes up heads in exactly  $3$  out of  $5$  flips. Find  $i+j$ .

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