
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]


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Given the function $f(x)=ax\sin x+\cos x$, and the slope of the tangent line to $f(x)$ at $x=\dfrac{\pi}{4}$ is $\dfrac{\sqrt{2}\pi}{8}$.

$(1)$ Find the value of $a$, and discuss the monotonicity of $f(x)$ on $[-\pi,\pi]$;

$(2)$ Let the function $g(x)=\ln(mx+1)+\dfrac{1-x}{1+x}, x\geqslant 0$, where $m > 0$. If for any $x_1 \in [0,+\infty)$ there always exists $x_2 \in [0,\dfrac{\pi}{2}]$ such that $g(x_1) \geqslant f(x_2)$ holds, find the range of values for $m$.

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