
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 vectors $\overrightarrow{m} = (\sin A, \sin B)$, $\overrightarrow{n} = (\cos B, \cos A)$, and $\overrightarrow{m} \cdot \overrightarrow{n} = \sin 2C$, where $A$, $B$, and $C$ are the angles opposite to sides $a$, $b$, and $c$ of $\triangle ABC$ respectively.  
(1) Find the magnitude of angle $C$.  
(2) If $a+b=2$, and $D$ is the midpoint of side $AB$, find the minimum value of $|\overrightarrow{CD}|$.

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