
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
Let \((x_{A}, y_{A}), (x_{B}, y_{B})\) be the coordinates of points \(A\) and \(B\) on a parabola. Then \(y_{A} = x_{A}^2\) and \(y_{B} = x_{B}^2\). Consider the case where \(x_{A} < 0\) and \(x_{B} > 0\). According to the conditions, the lengths of segments \(AC\) and \(CB\) are \(2t\) and \(t\) respectively, so point \(A\) is above point \(B\). Since \(\triangle APC \sim \triangle ARB\), we have the proportion \(\frac{y_{A} - y_{B}}{y_{A} - 8} = \frac{2t + t}{2t} = \frac{3}{2}\), from which \(y_{A} = 24 - 2y_{B}\).

Further, \(\triangle CSB \sim \triangle ARB\), therefore,
\[
\frac{x_{B} - x_{A}}{x_{B}} = \frac{2t + t}{t} = \frac{3}{2}, \quad \text{then} \ x_{A} = -2x_{B} \Rightarrow x_{A}^{2} = 4x_{B}^{2} \Rightarrow y_{A} = 4y_{B}
\]

Finally, solving the system \(\left\{ \begin{array}{c} y_{A} = 24 - 2y_{B} \\ y_{A} = 4y_{B} \end{array} \right.\), we get 
\[
y_{B} = 4 \Rightarrow x_{B} = 2 \Rightarrow x_{A} = -4
\]

By symmetry, for \(x_{A} > 0, x_{B} < 0\), there is a solution \(x_{B} = -2, x_{A} = 4\).

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