
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

Define \( f(x) = f\left(x ; a_{0}, \cdots, a_{2021}, b_{1}, \cdots, b_{2021}\right) = a_{0} + \sum_{k=1}^{2021} \left( a_{k} \cos 2 k \pi + b_{k} \sin 2 k \pi \right) \) where \( a_{k}, b_{k} \in \mathbb{R} \), and \( f(x) \) satisfies the properties:
1) \( f(x) + f\left(x + \frac{1}{2}\right) = f(2x) \quad (\forall x \in \mathbb{R}) \),
2) If \( f\left(x; a_{0}, \cdots, a_{2021}, b_{1}, \cdots, b_{2021}\right) = f\left(x; a_{0}', \cdots, a_{2021}', b_{1}', \cdots, b_{2021}' \right) \quad (\forall x \in \mathbb{R})\), then \( \left(a_{0}, \cdots, a_{2021}, b_{1}, \cdots, b_{2021}\right) = \left(a_{0}', \cdots, a_{2021}', b_{1}', \cdots, b_{2021}' \right) \).

Find all arrays \(\left(a_{0}, \cdots, a_{2021}, b_{1}, \cdots, b_{2021}\right)\) that satisfy the conditions.

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