
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 a minor axis length of $2$ and an eccentricity of $\frac{\sqrt{2}}{2}$, let line $l$ pass through the right focus and intersect the ellipse $C$ at two distinct points $A$ and $B$. Draw lines $AP$ and $BQ$ perpendicular to the line $x=2$ with foot points $P$ and $Q$ respectively. Denote $\lambda = \frac{AP+BQ}{PQ}$. If the slope of line $l$ is $k \geq \sqrt{3}$, then the range of $\lambda$ is _______ .

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