
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 randomly from all rational numbers in the interval \([0,3)\) that can be expressed as fractions \( \frac{n}{d} \) where \( n \) and \( d \) are integers with \( 1 \leq d \leq 4 \). What is the probability that \((\cos(a\pi) + i\sin(b\pi))^8 = 1\)?

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