
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 that the point $(\dfrac{5\pi }{12},2)$ is on the graph of the function $f(x)=2\sin (\omega x+\varphi )$ $( \omega > 0,0 < |\varphi | < \dfrac{\pi }{2} )$, and the lines $x=x_{1}$, $x=x_{2}$ are any two axes of symmetry for the graph of $y=f(x)$, with the minimum value of $|x_{1}-x_{2}|$ being $\dfrac{\pi }{2}$.

$(1)$ Find the interval of monotonic increase for the function $f(x)$ and the coordinates of the center of symmetry of its graph;

$(2)$ Let $A=\{x|\dfrac{\pi }{4}\leqslant x\leqslant \dfrac{\pi }{2}\}$, $B=\{x||f(x)-m| < 1\}$. If $A\subseteq B$, find the range of values for the real number $m$.

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