
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 $\sin \left( \frac{\pi }{3}-\alpha \right)=\frac{1}{2}$, find the value of $\cos \left( \frac{\pi }{6}+\alpha \right)$;

(2) Given $\cos \left( \frac{5\pi }{12}+\alpha \right)=\frac{1}{3}$ and $-\pi < \alpha < -\frac{\pi }{2}$, find the value of $\cos \left( \frac{7\pi }{12}-\alpha \right)+\sin \left( \alpha -\frac{7\pi }{12} \right)$:

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