
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 $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ with $a > 0$ and an eccentricity of $\frac{\sqrt{2}}{2}$, its right focus is $F(1, )$. A line $l$ is drawn through $F$ intersecting the ellipse $C$ at points $A$ and $B$, where $|A|=|FB|$. Let $T(, )$ be a point on the line segment $AB$. Find the range of values for the mid-length of $AT$ on $AB$ when $1 \leq \lambda \leq 2$.

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