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

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