
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 that $\sin \alpha - \cos \alpha = \frac{1}{5}$ ($\alpha$ is an angle in the third quadrant), find the values of $\sin \alpha \cdot \cos \alpha$ and $\sin \alpha + \cos \alpha$.
(2) Given that $\cos (40^{\circ}+x) = \frac{1}{4}$, where $-180^{\circ} < x < -90^{\circ}$, find the value of $\cos (140^{\circ}-x) + \cos ^{2}(50^{\circ}-x)$.

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