
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
In the $xOy$ plane, draw two lines $A_{1}B_{1}$ and $A_{2}B_{2}$ through a point $P$ on the $x$-axis, where the points $A_{1}$, $B_{1}$, $A_{2}$, and $B_{2}$ are all on the parabola $\Gamma: y^{2}=x$. Let $A_{1}B_{2}$ and $A_{2}B_{1}$ respectively pass through points $S$ and $T$ on the $x$-axis. Compare the magnitudes of $\overrightarrow{OS} \cdot \overrightarrow{OT}$ and $|OP|^{2}$.

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