
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
(1) Given the function $f(x)=\cos x+\sqrt{3}\sin x$, find the value of $f'(\frac{\pi}{3})$.

(2) In the polar coordinate system, the chord length intercepted by the line $\rho\sin \theta+\rho\cos \theta=2\sqrt{2}$ on the circle $\rho=2\sqrt{2}$ is \_\_\_\_\_\_.

(3) Given the function $f(x)=x^{3}+ax^{2}+bx-a^{2}-7a$ that attains a maximum value of $10$ at $x=1$, find the value of $a+b$.

(4) Given the function $f(x)=x^{3}-ax^{2}+3ax+1$ that has both a maximum and a minimum value in the interval $(-2,2)$, find the range of the real number $a$.

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