
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 $f\left( x \right)=\sqrt{3}\sin 2x+2{{\cos }^{2}}x+a$ defined on $\left[0,\dfrac{\pi }{2}\right]$ with a maximum value of 4.

(1) If $f(x)=m$ has two unequal real roots $x_{1}$ and $x_{2}$ in $\left[0,\dfrac{\pi }{2}\right]$, find the range of $m$ and the value of $x_{1}+x_{2}$.

(2) Consider an acute triangle $\triangle ABC$ with angles $A$, $B$, $C$. If $f\left( A- \dfrac{\pi }{6} \right)=4$, $BC=\sqrt{7}$, and $\sin B=\dfrac{\sqrt{21}}{7}$, find the length of $AC$ and the area of $\triangle ABC$.

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