
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 line $l$: $y=k(x-1)$ intersects the $x$-axis at point $A$, the $y$-axis at point $B$, and line $y=x$ at point $C$,
(1) If $k=3$, find the value of $\frac{|BC|}{|AC|}$;
(2) If $|BC|=2|AC|$, find the equation of line $l$.

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