
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 the function \( y = f(x) \) satisfy: for all \( x \in \mathbb{R} \), \( y = f(x) \geqslant 0 \), and \( f(x+1) = \sqrt{9 - f(x)^2} \). When \( x \in [0,1) \),
$$
f(x) = \begin{cases} 
2^x, & 0 \leqslant x < \frac{1}{2}, \\
\log_{10} (x + 31), & \frac{1}{2} \leqslant x < 1
\end{cases}
$$

Find \( f(\sqrt{1000}) \).

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