
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 $f(x)$ be an increasing function defined on $(0, +\infty)$ satisfying $f(xy) = f(x)f(y)$ for all positive real numbers $x$ and $y$, and $f(2) = 4$.

(1) Find the values of $f(1)$ and $f(8)$.
(2) Solve the inequality for $x$: $16f\left(\frac{1}{x-3}\right) \geq f(2x+1)$.

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