
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 $$f(x)=\cos x( \sqrt {3}\sin x+\cos^{3}x)+\sin x( \sqrt {3}\cos x-\sin^{3}x)$$  
(1) Find the interval of monotonic increase for $f(x)$;  
(2) Let $\triangle ABC$ have internal angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively. If $a^{2}+c^{2}=ac+b^{2}$, and $f(A)=0$, with $b+c= \sqrt {2}+ \sqrt {3}$, find the values of $b$ and $c$.

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