
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
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. The vectors $\overrightarrow{m}=(a+c,b-a)$, $\overrightarrow{n}=(a-c,b)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.

(1) Find the measure of angle $C$;

(2) If vectors $\overrightarrow{s}=(0,-1)$, $\overrightarrow{t}=(\cos A, 2\cos^{2}\frac{B}{2})$, find the range of $|\overrightarrow{s}+\overrightarrow{t}|$.

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