
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 $C(x)$ and $S(x)$ be periodic functions defined as follows:

$$
\begin{aligned}
& C(x)=\left\{\begin{array}{lll}
1-x & \text{if} & 0 \leq x \leq 2 \\
x-3 & \text{if} & 2 < x \leq 4
\end{array}\right. \\
& C(x)=C(x+4), \quad S(x)=C(x-1)
\end{aligned}
$$

Are the following relations true?

$$
\begin{gathered}
C(2x) = [C(x)]^2 - [S(x)]^2 \\
S(2x) = 2 C(x) S(x)
\end{gathered}
$$

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