
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
Given a parabola \( y = x^2 \), a line with a slope of 1 passing through the origin \( O \) intersects the parabola at point \( P_1 \). A line with a slope of \( \frac{1}{2} \) passing through point \( P_1 \) intersects the parabola at point \( P_2 \). Another line with a slope of \( \frac{1}{4} \) passing through point \( P_2 \) intersects the parabola at point \( P_3 \), and so on. A line with a slope of \( \frac{1}{2^n} \) passing through point \( P_n \) intersects the parabola at point \( P_{n+1} \), and so forth. Let the coordinates of point \( P_n \) be \( (x_n, y_n) \). Find \( x_{2n+1} \), and determine the point that \( P_1, P_3, P_5, \cdots \) approach.

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