
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
a. Determine all points \( P(x, y) \) such that \((0,0)\), \((1,1)\), \((1,0)\), and \(P\) are vertices of a parallelogram.

b. Two parallel lines intersect the (horizontal) parabola \( x = y^{2} \) at four distinct points: \((0,0)\), \((1,1)\), \((9,3)\), and \(Q\). Determine all possible coordinates of the point \( Q \).

c. Two parallel lines intersect the parabola \( x = y^{2} \) at four distinct points: \((0,0)\), \((1,1)\), \(\left(a^{2}, a\right)\), and \(V\). Here \( a \neq 0, \pm 1 \) is a real number. Determine all possible coordinates of the point \( V \). The answer should be expressed in terms of \( a \).

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