
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 ellipse \(Γ\): \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (\(a > b > 0\)), two lines passing through the origin, \(l_1\) and \(l_2\), intersect \(Γ\) at points \(A\), \(B\), and \(C\), \(D\) respectively, forming a parallelogram \(ACBD\).
1. When \(ACBD\) is a square, find the area \(S\) of the square.
2. If lines \(l_1\) and \(l_2\) are symmetric with respect to the \(y\)-axis and for any point \(P\) on \(Γ\), the distances to \(l_1\) and \(l_2\) are \(d_1\) and \(d_2\) respectively, find the slopes of \(l_1\) and \(l_2\) when the value of \(d_1^2+d_2^2\) is constant.
3. When \(ACBD\) is a rhombus, and the circle \(x^2+y^2=1\) is inscribed inside the rhombus \(ACBD\), find the relationship between \(a\) and \(b\).

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