
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 \( C \) defined by \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with \(a > b > 0\) and an eccentricity of \(\frac{2}{3}\). Points \(A_{1}\) and \(A_{2}\) are the left and right vertices of the ellipse \(C\), \(B\) is the top vertex, and \(F_{1}\) is the left focus of the ellipse \(C\). The area of triangle \(\triangle A_{1} F_{1} B\) is \(\frac{\sqrt{5}}{2}\).

(1) Find the equation of the ellipse \(C\).

(2) Suppose a moving line \(l\) passes through the point \(D(1,0)\) and intersects the ellipse \(C\) at points \(E\) and \(F\) (point \(E\) is above the x-axis). Points \(M\) and \(N\) are the intersections of lines \(A_{1} E\) and \(A_{2} F\) with the y-axis, respectively. Find the value of \(\frac{|OM|}{|ON|}\).

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