
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 $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. The vectors $\overrightarrow{m} = (a-b,\sin A+\sin C)$ and $\overrightarrow{n} = (a-c, \sin(A+C))$ are collinear.  
(1) Find the value of angle $C$;  
(2) If $\overrightarrow{AC} \cdot \overrightarrow{CB} = -27$, find the minimum value of $|\overrightarrow{AB}|$.

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