
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 an ellipse \( E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) \((a > b > 0)\), with a minor axis length of \( 2 \sqrt{3} \) and an eccentricity of \( e_{1} \); a hyperbola \(\frac{x^{2}}{m} - \frac{y^{2}}{n} = 1 \) \((m, n > 0) \) whose asymptotes are \( y = \pm \sqrt{3} x \), and an eccentricity of \( e_{2} \). It is given that \( e_{1} e_{2} = 1 \).

(1) Find the equation of the ellipse \(E\);

(2) If \( A \) is the right vertex of the ellipse \( E \), \( P\left(-1, \frac{3}{2}\right) \), and the line \( AP \) intersects the y-axis at point \( H \). Another line through point \( H \) intersects the ellipse \( E \) at points \( M \) and \( N \). Let the area of \( \triangle HMA \) be \( S_{1} \) and the area of \( \triangle PHN \) be \( S_{2} \), where \( S_{1} = 6 S_{2} \). Find the equation of the line \( MN \).

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