
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
The vertex of a parabola is at the origin, and its focus is on the positive half of the $x$-axis. The line $x+y-1=0$ intersects the parabola at points $A$ and $B$, and $|AB|= \dfrac {8 \sqrt {6}}{11}$.  
$(1)$ Find the equation of the parabola;  
$(2)$ Is there a point $C$ on the $x$-axis such that $\triangle ABC$ is an equilateral triangle? If it exists, find the coordinates of point $C$; if not, explain why.

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