
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 $f(x)=5 \sqrt{3}\sin (\pi-x)+5\sin \left( \dfrac{\pi}{2}+x\right)+5$,
$(1)$ find the axis of symmetry and the center of symmetry for the function $f(x)$;
$(2)$ The graph of function $f(x)$ is translated to the right by $\dfrac{\pi}{6}$ units, then translated downwards by $a(a > 0)$ units, and then the horizontal coordinate is compressed to half of its original length to obtain the graph of function $g(x)$. If the maximum value of function $g(x)$ is $2$, find the analytic expression for $g(x)$ and its intervals of monotonicity.

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