
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 $180^{\circ} < \alpha < 360^{\circ}$, find the value of $\sqrt{1 + \cos \alpha}$ and select the correct option:
A) $-\sqrt{2}\cos\frac{\alpha}{2}$
B) $\sqrt{2}\cos\frac{\alpha}{2}$
C) $-\sqrt{2}\sin\frac{\alpha}{2}$
D) $\sqrt{2}\sin\frac{\alpha}{2}$

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