
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
At a concert  $10$  singers will perform. For each singer  $x$ , either there is a singer  $y$  such that  $x$  wishes to perform right after  $y$ , or  $x$  has no preferences at all. Suppose that there are  $n$  ways to order the singers such that no singer has an unsatisfied preference, and let  $p$  be the product of all possible nonzero values of  $n$ . Compute the largest nonnegative integer  $k$  such that  $2^k$  divides  $p$ .

*Proposed by Gopal Goel*

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