
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 $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ with an eccentricity of $\frac{1}{2}$, points $F\_1$, $F\_2$ are the left and right foci of the ellipse $E$, respectively. A moving line $l$ passes through the fixed point $Q(0,2)$ and intersects the ellipse $E$ at points $A$ and $B$. When $F\_1$, $A$, and $B$ are collinear, the perimeter of $\triangle F\_2AB$ is $8$.
(1) Find the standard equation of the ellipse $E$;
(2) Let $D$ be the midpoint of the chord $AB$, and $E(0,t)$ is a point on the $y$-axis such that $DE \perp AB$, find the range of values for $t$.

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