
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)= \sqrt{3}\sin 2x+2{\cos }^{2}x+m $ on $x\in[0, \frac{π}{2}] $ with a maximum value of $6$.

(1) Find the value of $m$ and the interval where the function $f(x)$ is monotonically increasing;

(2) In $\triangle ABC$, angles $A$, $B$, $C$ have opposite sides $a$, $b$, $c$ respectively, with $f(A)=5$, $a=4$, and the area of $\triangle ABC$ being $ \sqrt{3} $, find the value of $b+c$.

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