
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
Consider a square \(A B C D\) with center \(O\). Let \(E, F, G,\) and \(H\) be points on the interior of sides \(AB, BC, CD,\) and \(DA\), respectively, such that \(AE = BF = CG = DH\). It is known that \(OA\) intersects \(HE\) at point \(X\), \(OB\) intersects \(EF\) at point \(Y\), \(OC\) intersects \(FG\) at point \(Z\), and \(OD\) intersects \(GH\) at point \(W\). Let \(x\) and \(y\) be the lengths of \(AE\) and \(AH\), respectively.

a) Given that the Area of \(EFGH\) is \(1 \text{ cm}^2\), calculate the value of \(x^2 + y^2\).

b) Verify that \(HX = \frac{y}{x+y}\). Then, conclude that \(X, Y, Z,\) and \(W\) are the vertices of a square.

c) Calculate the product:

    \[\text{Area}(ABCD) \cdot \text{Area}(XYZW).\]



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