
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 that an equilateral triangle \( \triangle ABC \) is inscribed in the parabola \( y = x^2 - \frac{71}{36} \), and point \( P \) satisfies \( \overrightarrow{P A} + \overrightarrow{P B} + \overrightarrow{P C} = \overrightarrow{0} \). Does there exist a fixed point \( Q \) such that the distance from point \( P \) to point \( Q \) is equal to the distance from point \( P \) to the \( x \)-axis? If such a point exists, find the coordinates of point \( Q \); if it does not exist, explain why.

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