
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 the function $f(x)$ such that $f(xy) = f(x)f(y)$ for any real numbers $x$ and $y$, and $f(-1) = 1$, $f(27) = 9$. When $0 \leq x < 1$, it holds that $0 \leq f(x) < 1$.
1. Find the values of $f(0)$ and $f(3)$;
2. Determine the parity of $f(x)$;
3. Determine the monotonicity of $f(x)$ on the interval $[0, +\infty)$, and provide a proof;
4. If $a \geq 0$ and $f(a+1) \leq 39$, find the range of values for $a$.

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