
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 $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, there is a point $M(2,1)$ inside it. Two lines $l_1$ and $l_2$ passing through $M$ intersect the ellipse $E$ at points $A$, $C$ and $B$, $D$ respectively, and satisfy $\overrightarrow{AM}=\lambda \overrightarrow{MC}, \overrightarrow{BM}=\lambda \overrightarrow{MD}$ (where $\lambda > 0$, and $\lambda \neq 1$). If the slope of $AB$ is always $- \frac{1}{2}$ when $\lambda$ changes, find the eccentricity of the ellipse $E$.

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