
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 a triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively. Also given are vectors $\vec{m} = (a+c, b-a)$ and $\vec{n} = (a-c, b)$, where $\vec{m} \perp \vec{n}$.

(I) Find the measure of angle $C$.
(II) If vectors $\vec{s} = (0, -1)$ and $\vec{t} = (\cos A, 2\cos^2 \frac{B}{2})$, find the range of values for $|\vec{s} + \vec{t}|$.

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