
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 the curve \( x y^{2} = 1 (y > 0) \), the slope of the tangent line at a point \((x_0, y_0)\) on the curve is \( k = -\frac{1}{2 \sqrt{x_0^{3}}} \). A line parallel to the \( y \)-axis through the point \( P_1 (1, 0) \) intersects the curve at \( Q_1 \). The tangent line to the curve at \( Q_1 \) intersects the \( x \)-axis at \( P_2 \). A line parallel to the \( y \)-axis through \( P_2 \) intersects the curve at \( Q_2 \). This process is repeated, generating a sequence of points \( P_1, P_2, \cdots \) and \( Q_1, Q_2, \cdots \). Let \( l_n = \left| P_n Q_n \right| \). Find the value of \( l_{2003} \).

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