
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 opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and it satisfies $b\cos C+c\cos B=2a\cos B$.  
$(1)$ Find the magnitude of angle $B$;  
$(2)$ If $b= \sqrt {3}$, find the maximum value of $a^{2}+c^{2}$.

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