
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 that in $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$, and 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}
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