
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
1) Consider an arbitrary line \( y = k - b \) that intersects a parabola at points \( A \) and \( B \). The x-coordinates of the endpoints of chord \( AB \) must satisfy the equation \( kx - b = x^2 \), i.e., \( x^2 - kx + b = 0 \). The x-coordinates are \( x_A = \frac{k - \sqrt{k^2 - 4b}}{2} \) and \( x_B = \frac{k + \sqrt{k^2 - 4b}}{2} \). The x-coordinate \( x_C \) of the midpoint of the segment \( AB \) is \( x_C = \frac{x_A + x_B}{2} = \frac{k}{2} \) and depends only on the slope \( k \), not on \( b \). Therefore, if two chords \( A1B1 \) and \( A2B2 \) are parallel, their midpoints \( C1 \) and \( C2 \) have the same x-coordinates and the line \( C1C2 \) is parallel to the \( OY \) axis. The perpendicular chord \( A0B0 \) to this line is parallel to the \( OX \) axis, and due to symmetry, its midpoint lies on the \( OY \) axis. Therefore, the line through point \( C0 \) parallel to \( C1C2 \) is the \( OY \) axis. It intersects the parabola at the vertex \( O \), and the line passing through point \( O \) perpendicular to \( OY \) will be the \( OX \) axis.

2) Let \( t = \frac{1}{x} \); hence, \( x = \frac{1}{t} \). Then we get the equality: \( f\left( \frac{1}{t} \right) + \frac{5}{t} f(t) = \frac{3}{t^3} \). Because this equality is true for any \( t \neq 0 \), it is true for \( t = x \). Thus we have: \( f\left( \frac{1}{x} \right) + \frac{5}{x} f(x) = \frac{3}{x^3} \). Then we obtain the system of equalities:
\[
\begin{cases}
f\left( \frac{1}{x} \right) + \frac{5}{x} f(x) = \frac{3}{x^3} \\
f(x) + 5x f\left( \frac{1}{x} \right) = 3x^3
\end{cases}
\]
or equivalently:
\[
\begin{cases}
5x f\left( \frac{1}{x} \right) + 25 f(x) = \frac{15}{x^2} \\
5x f\left( \frac{1}{x} \right) + f(x) = 3x^3
\end{cases}
\]
Subtracting the second equation from the first, we get \( 24 f(x) = \frac{15}{x^2} - 3x^3 \), hence \( f(x) = \frac{5}{8x^2} - \frac{x^3}{8} \).

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