
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
How far is the secant line passing through the points $\left(n, n^{2}\right)$ and $\left(n+1,(n+1)^{2}\right)$ on the $y=x^{2}$ parabola from the tangent line of the parabola parallel to it? In what ratio does the projection of the tangent point divide the chord formed by the secant line?

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