
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 $ABC$, let the sides opposite angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. It is known that $a^{2}+c^{2}-b^{2}=ac$ and $\sqrt{2}b= \sqrt{3}c$.
$(1)$ Find the measure of angle $A$.
$(2)$ Let $f(x)=1+\cos(2x+B)-\cos(2x)$. Find the maximum value of the function $f(x)$.

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