
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

A pair of functions \( f(x) \) and \( g(x) \) is called a Payneful pair if:
1. \( f(x) \) is a real number for all real numbers \( x \),
2. \( g(x) \) is a real number for all real numbers \( x \),
3. \( f(x+y) = f(x) g(y) + g(x) f(y) \) for all real numbers \( x \) and \( y \),
4. \( g(x+y) = g(x) g(y) - f(x) f(y) \) for all real numbers \( x \) and \( y \),
5. \( f(a) \neq 0 \) for some real number \( a \).

For every Payneful pair of functions \( f(x) \) and \( g(x) \):

(a) Determine the values of \( f(0) \) and \( g(0) \).

(b) If \( h(x) = (f(x))^2 + (g(x))^2 \) for all real numbers \( x \), determine the value of \( h(5)h(-5) \).

(c) If \( -10 \leq f(x) \leq 10 \) and \( -10 \leq g(x) \leq 10 \) for all real numbers \( x \), determine the value of \( h(2021) \).

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