
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
Determine for what values of \( y \) and \( x \) the inequality \( y + x > |\frac{x}{2}| \) holds.
**A)** \( y > -\frac{x}{2} \text{ for all values of } x \)  
**B)** \( y > -x \text{ for all values of } x \)  
**C)** \( y > -\frac{x}{2} \text{ for } x \geq 0 \text{ and } y > -\frac{3x}{2} \text{ for } x < 0 \)  
**D)** \( y < -x \text{ for } x \geq 0 \text{ and } y < \frac{3x}{2} \text{ for } x < 0 \)

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