
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 $f(x)=-2\cos x-x$, $g(x)=\ln x+\frac{k}{x}$.

$(1)$ Find the interval of monotonic increase for the function $f(x)$;

$(2)$ If for $\forall x_1 \in [0,\frac{1}{2}]$, $\forall x_2 \in [\frac{1}{2},1]$, it holds that $f(x_1) < g(x_2)$, find the range of the real number $k$.

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