
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) Calculate: $\lg 4+\lg 9+2\sqrt{{(\lg 6)^2}-\lg 36+1}=$\_\_\_\_\_\_\_\_\_\_\_\_\_

(2) If the terminal sides of angles $\alpha$ and $\beta$ are symmetric about the line $x+y=0$, and $\alpha =-\frac{\pi }{3}$, then the set of angle $\beta$ is\_\_\_\_\_\_\_\_\_\_\_\_\_ 

(3) The perimeter of sector $AOB$ is $8$, and the central angle is $2$ radians, then its area is\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

(4) Given $\sin \left( \frac{\pi}{6}+\alpha\right)= \frac{3}{5}$, $\frac{\pi}{3} < \alpha < \frac{5\pi}{6} $, then the value of $\cos \alpha$ is\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

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