
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 $f(x)=\log^{\frac{1}{2}}(x^2-mx-m)$,

(1) If $m=1$, find the domain of the function $f(x)$; (2) If the range of the function $f(x)$ is $\mathbb{R}$, find the range of the real number $m$;

(3) If the function $f(x)$ is increasing in the interval $(-\infty,1- \sqrt{3})$, find the range of the real number $m$.

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