
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 *permutation* of a finite set is a one-to-one function from the set onto itself.  A *cycle* in a permutation  $P$  is a nonempty sequence of distinct items  $x_1$ ,  $\ldots\,$ ,  $x_n$  such that  $P(x_1) = x_2$ ,  $P(x_2) = x_3$ ,  $\ldots\,$ ,  $P(x_n) = x_1$ .  Note that we allow the 1-cycle  $x_1$  where  $P(x_1) = x_1$  and the 2-cycle  $x_1, x_2$  where  $P(x_1) = x_2$  and  $P(x_2) = x_1$ .  Every permutation of a finite set splits the set into a finite number of disjoint cycles.  If this number equals 2, then the permutation is called *bi-cyclic*.  Compute the number of bi-cyclic permutations of the 7-element set formed by the letters of "PROBLEM".

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