
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
The total score for this problem is 12 points: part (I) is worth 7 points and part (II) is worth 5 points.

Given that $O$ is the origin of the coordinate system, vector $\overrightarrow{a} = (\sin\alpha, 1)$, $\overrightarrow{b} = (\cos\alpha, 0)$, and $\overrightarrow{c} = (-\sin\alpha, 2)$. Point $P$ lies on line segment $AB$ such that point $B$ divides the directed line segment in the ratio 1:1.

(1) Define the function $f(\alpha) = \overrightarrow{OP} \cdot \overrightarrow{c}$, where $\alpha \in \mathbb{R}$. Discuss the monotonicity of the function $f(\alpha)$ and find its range.

(2) If points $O$, $P$, and $C$ are collinear, find the value of $|\overrightarrow{OB} + \overrightarrow{OP}|$.

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