
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 $\triangle ABC$ with vertex $C$ on the line $3x - y = 0$, and vertices $A$ and $B$ have coordinates $(4, 2)$ and $(0, 5)$ respectively.
(Ⅰ) Find the equation of the line that passes through point A and has equal intercepts on the x-axis and y-axis;
(Ⅱ) If the area of $\triangle ABC$ is $10$, find the coordinates of vertex $C$.

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