
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 $F_{1}(-1,0)$, its eccentricity $e$, and $b$ being the geometric mean of $3e$ and $a$.

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

(2) A line with a slope angle $α$ passes through the origin $O$ and intersects $C$ at points $A$ and $B$. Another line with a slope angle $β$ passes through $F_{1}$ and intersects $C$ at points $D$ and $E$. If $α+β=π$, find the value of $\frac{|AB|^{2}}{|DE|}$.

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