
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 $Q$: $\frac{x^2}{a^2} + y^2 = 1 (a > 1)$, $F\_1$ and $F\_2$ are its left and right foci, respectively. A circle with the segment $F\_1F\_2$ as its diameter intersects the ellipse $Q$ at exactly two points.
(1) Find the equation of the ellipse $Q$;
(2) Let line $l$ pass through point $F\_1$ and not be perpendicular to the coordinate axes, intersecting the ellipse at points $A$ and $B$. The perpendicular bisector of segment $AB$ intersects the $x$-axis at point $P$, and the $x$-coordinate of point $P$ has a value range of $\left[- \frac{1}{4}, 0\right)$. Find the minimum value of $|AB|$.

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