
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
Given that rational numbers  $a$  and  $b$  are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$  where $n$ and $d$ are integers with $1\leq d\leq 5$, determine the probability that $(\cos(a\pi)+i\sin(b\pi))^4$ is a real number.

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