
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 vertices of $\triangle ABC$ have the Cartesian coordinates $A(3,4)$, $B(0,0)$, and $C(c,0)$.
$(1)$ If $\overrightarrow{AB} \cdot \overrightarrow{AC} = 0$, find the value of $c$;     
$(2)$ If $c=5$, find the value of $\sin A$.

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