
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
$(1)$ Given: There exists a certain correspondence $f$ between $x$ and $y$ in the table, where $y = f(x)$ (where $x \geqslant 0$, $y \geqslant 0$).<br/>

\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & \ldots & 0.0001 & 0.01 & 1 & 100 & 10000 & \ldots \\
\hline
y & \ldots & a & 0.1 & 1 & 10 & b & \ldots \\
\hline
\end{array}
\]

Then $a =$ ______, $b =$ ______.<br/>

$(2)$ According to the correspondence $f$ in $(1)$, fill in the blanks:<br/>

If $f(10) \approx 3.16$, then $f(1000) \approx$ ______.<br/>
If $f(3.68) \approx 1.918$, then $f($ ______ $) \approx 191.8$.

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