
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 $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, its left focus is $F_1$.
1. If the left focus $F_1$ and the two endpoints of the minor axis of the ellipse $E$ form the vertices of an equilateral triangle, and the point $Q(\sqrt{3}, \frac{1}{2})$ lies on the ellipse $E$. Find the equation of the ellipse $E$.
2. A line $l_1$ passing through the origin with a slope of $t$ ($t > 0$) intersects the ellipse $E$ from (1) at two distinct points $G$ and $H$. Let $B_1(0, 1)$, $A_1(2, 0)$. Find the equation of the line $l_1$ when the area of the quadrilateral $A_1GB_1H$ is maximized.
3. A line $l_2$ passing through the left focus $F_1$ intersects the ellipse $E$ at points $M$ and $N$. The line $l_2$ intersects the line $x = -p$ ($p > 0$) at point $P$, where $p$ is a constant. Let $\overrightarrow{PM} = \lambda \overrightarrow{MF_1}$, $\overrightarrow{PN} = \mu \overrightarrow{NF_1}$. Calculate the value of $\lambda + \mu$ (expressed in terms of $p$, $a$, and $b$).

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