
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 $C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ (where $a > b > 0$) with its left focus $F_1(-1, 0)$, and the eccentricity of $C$ is $e$, with $b$ as the geometric mean between $3e$ and $a$:
(1) Find the equation of the curve $C$;
(2) A line with slope angle $\alpha$ passing through the origin $O$ intersects $C$ at points $A$ and $B$, a line with slope angle $\beta$ passing through $F_1$ intersects $C$ at points $D$ and $E$, if $\alpha + \beta = \pi$, find the value of $\frac{|AB|^2}{|DE|}$.

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