
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 parabola $E$: $y=ax^{2}$ with three distinct points $A(1,1)$, $B$, and $C$ that satisfy the relationship $\overrightarrow{AB}\cdot \overrightarrow{BC}=0$.
(1) Find the equation of the parabola $E$;
(2) Find the minimum area of the circumcircle of $\triangle ABC$ and the equation of the circumcircle in this case.

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