
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 that angle $\alpha$ is in the fourth quadrant, and the x-coordinate of the point where its terminal side intersects the unit circle is $\frac{1}{3}$.
(1) Find the value of $\tan \alpha$;
(2) Find the value of $\frac{\sin^{2}\alpha - \sqrt{2}\sin \alpha\cos \alpha}{1 + \cos^{2}\alpha}$.

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