
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 the coordinates of the three vertices of $\triangle ABC$ are $A(-1, 0)$, $B(4, 0)$, and $C(0, c)$.
(1) If $\overrightarrow{AC} \perp \overrightarrow{BC}$, find the value of $c$;
(2) When $c$ satisfies the conclusion of question (1), find the coordinates of the centroid $G(x, y)$ of $\triangle ABC$.

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