
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 that $F_1$ and $F_2$ are the left and right foci of the ellipse $E$, respectively, and point $P\left(\begin{matrix} 1, \frac{3}{2} \end{matrix}\right)$ is a point on it, satisfying $|PF_1|+|PF_2|=4$.

$(1)$ 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 $l_1$ and 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 the quadrilateral $ABCD$.

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