
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 $\Gamma$: $\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1(a>b>0)$, there is a point $P(1,1)$ inside the ellipse. Two lines $l_{1}$ and $l_{2}$ passing through point $P$ intersect the ellipse $\Gamma$ at points $A$, $C$ and $B$, $D$ respectively, satisfying $\overrightarrow{AP}=λ\overrightarrow{PC}$, $\overrightarrow{BP}=λ\overrightarrow{PD}$. If the slope of line $CD$ is always $-\frac{1}{4}$ as $\lambda$ varies, find the eccentricity of the ellipse $\Gamma$.

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