
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 line $l$: $y=kx$ intersects with circle $C_{1}$: $(x-1)^{2}+y^{2}=1$ at points $A$ and $B$, circle $C_{2}$ is externally tangent to circle $C_{1}$ and is tangent to line $l$ at point $M(3, \sqrt {3})$, find:
1. The value of $k$
2. The value of $|AB|$
3. The equation of circle $C_{2}$

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