
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 vectors $m=(\sin x- \sqrt {3}\cos x,1)$, $n=(\sin ( \frac {π}{2}+x), \frac { \sqrt {3}}{2})$, if $f(x)=m\cdot n$.

$(I)$ Find the interval of monotonic increase for $f(x)$;

$(II)$ Given that in $\triangle ABC$, the angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively, and $a=3$, $f( \frac {A}{2}+ \frac {π}{12})= \frac {1}{2}$, $\sin C=2\sin B$, find the values of $A$, $c$, $b$.

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