
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 the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with its left focus $F_1(-1,0)$, the eccentricity of $C$ is $e$, and $b$ is the geometric mean of $3e$ and $a$.

$(1)$ Find the equation of curve $C$;

$(2)$ A line passing through the origin $O$ with an inclination angle of $\alpha$ intersects $C$ at points $A$ and $B$, and a line passing through $F_1$ with an inclination angle of $\beta$ 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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