
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
Let  $a_1$ ,  $a_2$ ,  $a_3$ ,  $a_4$ , and  $a_5$  be random integers chosen independently and uniformly from the set  $\{ 0, 1, 2, \dots, 23 \}$ .  (Note that the integers are not necessarily distinct.)  Find the probability that
\[
  \sum_{k=1}^{5} \operatorname{cis} \Bigl( \frac{a_k \pi}{12} \Bigr) = 0.
\]
(Here  $\operatorname{cis} \theta$  means  $\cos \theta + i \sin \theta$ .)



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