
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_n $  be the product of the complex roots of  $ x^{2n} = 1 $  that are in the first quadrant of the complex plane. That is, roots of the form  $ a + bi $  where  $ a, b > 0 $ . Let  $ r = a_1 \cdots a_2 \cdot \ldots \cdot a_{10} $ . Find the smallest integer  $ k $  such that  $ r $  is a root of  $ x^k = 1 $ .

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