
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 $f(x)=\lg (x+1)$.

$(1)$ If $h(x)= \sqrt{f(x)}$, find the domain of $h(x)$;

$(2)$ If $y=g(x)$ is an even function defined on the domain $[-1,1]$ and when $0 \leqslant x \leqslant 1$, $g(x)=f(x)$, find the expression for $y=g(x)$.

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