
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 parabola $C: y=x^{2}$, and a point $P(0,2)$. Points $A$ and $B$ lie on the parabola, and the distance from point $P$ to line $AB$ is 1.  
(1) If the inclination angle of line $AB$ is $\frac{\pi}{3}$, find the equation of line $AB$.  
(2) Find the minimum value of $|AB|$.

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