
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
Among all pairs of real numbers $(x, y)$ such that $\cos \cos x = \cos \cos y$ with $-5 \pi \le x, y \le 5 \pi$, calculate the probability that $X = Y$, where $(X, Y)$ is a randomly selected pair.

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