
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 $\Gamma: \frac{x^2}{4} + \frac{y^2}{2} = 1$, two distinct lines $l_1$ and $l_2$ with complementary sloping angles are drawn through point $P(1,1)$. Let line $l_1$ intersect the ellipse $\Gamma$ at points $A$ and $B$, and let line $l_2$ intersect the ellipse at points $C$ and $D$.
$(1)$ If $P(1,1)$ is the midpoint of the segment $AB$, find the equation of line $AB$.
$(2)$ Let $\lambda= \frac{|AB|}{|CD|}$, determine the range of $\lambda$.

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