
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
A. Given $F_1$ and $F_2$ are the left and right foci of the ellipse $E$, and point $P(1,\frac{3}{2})$ is a point on it, with $|PF_1| + |PF_2| = 4$.

(1) Find the standard equation of the ellipse $E$;

(2) A line $l_1$ passing through $F_1$ intersects the ellipse $E$ at points $A$ and $B$, and a line $l_2$ parallel to $l_1$ passing through $F_2$ intersects the ellipse $E$ at points $C$ and $D$. Find the maximum value of the area $S_{\text{quadrilateral ABCD}}$ of quadrilateral $ABCD$.

B. Given the function $f(x) = \frac{a\ln x}{x} + b$ ($a,b \in \mathbb{R}$), the equation of the tangent line to the graph of the function at point $(1,f(1))$ is $y = x - 1$.

(1) Find the values of real numbers $a$, $b$ and the monotonic intervals of the function $f(x)$;

(2) When $f(x_1) = f(x_2)$ ($x_1 \neq x_2$), compare the sum of $x_1 + x_2$ with $2e$ (where $e$ is the base of the natural logarithm).

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