
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 $ABC$ with vertex $A(2,1)$, vertex $B(-2,0)$, and the equation of the bisector line of angle $C$ as $x+y=0$.  
(1) Find the coordinates of vertex $C$.  
(2) Calculate the area of $\triangle ABC$.

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