
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 real numbers \( x \) for which \( x^{2} + 2x - 8 = 0 \).

(b) Determine the values of \( b \) and \( c \) for which the parabola with equation \( y = x^{2} + bx + c \) passes through the points \((1,2)\) and \((2,0)\).

(c) A ball is thrown from a window at the point \((0,2)\). The ball flies through the air along the parabola with equation \( y = a(x-1)^{2} + \frac{8}{3} \), for some real number \( a \). Determine the positive real number \( d \) for which the ball hits the ground at \((d, 0)\).

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