
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
Rational numbers \(a\) and \(b\) are chosen at random among all rational numbers in the interval \([0,2)\) that can be written as fractions \(\frac{n}{d}\) where \(n\) and \(d\) are integers with \(1 \leq d \leq 10\). What is the probability that \[(\cos(a\pi) + i\sin(b\pi))^6\] is a real number?

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