
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
In triangle $ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\dfrac{a}{\cos C \sin B} = \dfrac{b}{\sin B} + \dfrac{c}{\cos C}$.
$(1)$ Find the maximum value of $\sin(A+B) + \sin A \cos A + \cos(A-B)$;
$(2)$ If $b = \sqrt{2}$, find the perimeter of $\triangle ABC$ when its area is maximized.

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