
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 sides $a$, $b$, $c$ are opposite to angles $A$, $B$, $C$, respectively, and satisfy the equation $b\cos C=(2a+c)\cos(\pi-B)$.

$(1)$ Find the size of angle $B$;

$(2)$ If $b= \sqrt{13}$ and the area of $\triangle ABC$ is $\frac{3 \sqrt{3}}{4}$, find $a+c$.

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