
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 $g(x)= \frac {1}{x\cdot \sin \theta}+\ln x$ is increasing on $[1,+\infty)$ and $\theta\in(0,\pi)$, $f(x)=mx- \frac {m-1}{x}-\ln x$ $(m\in\mathbb{R})$  
$(1)$ Find the value of $\theta$;  
$(2)$ If $f(x)-g(x)$ is a monotonic function on $[1,+\infty)$, find the range of $m$.

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