
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 $\tan \alpha =\frac{1}{3}$, find the value of $\frac{1}{2\sin \alpha \cos \alpha +\cos^2 \alpha }$;

(2) Simplify $\frac{\tan (\pi -\alpha )\cos (2\pi -\alpha )\sin (-\alpha +\frac{3\pi }{2})}{\cos (-\alpha -\pi )\sin (-\pi -\alpha )}$.

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