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

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