
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 $g(x)=x\sin \theta-\ln x-\sin \theta$ is monotonically increasing on $[1,+\infty)$, where $\theta\in(0,\pi)$.

$(1)$ Find the value of $\theta$;

$(2)$ If $f(x)=g(x)+ \frac{2x-1}{2x^{2}}$, when $x\in[1,2]$, compare the size relationship between $f(x)$ and $f'(x)+ \frac{1}{2}$ (where $f'(x)$ is the derivative of $f(x)$), please write out the detailed reasoning process;

$(3)$ When $x > 0$, $e^{x}-x-1\geqslant kg(x+1)$ always holds, find the range of values for $k$.

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