
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
For some positive integer  $n$ , a coin will be ﬂipped  $n$  times to obtain a sequence of  $n$  heads and tails. For each ﬂip of the coin, there is probability  $p$  of obtaining a head and probability  $1-p$  of obtaining a tail, where  $0<p<1$  is a rational number.
Kim writes all  $2^n$  possible sequences of  $n$  heads and tails in two columns, with some sequences in the left column and the remaining sequences in the right column. Kim would like the sequence produced by the coin ﬂips to appear in the left column with probability  $1/2$ .
Determine all pairs  $(n,p)$  for which this is possible.

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