
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 $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$, its left focus is $F_{1}(-1,0)$, and $P$ is a vertex on the ellipse, with $\angle PF_{1}O=45^{\circ}$ ($O$ is the origin).
$(1)$ Find the values of $a$ and $b$;
$(2)$ It is known that the line $l_{1}: y=kx+m_{1}$ intersects the ellipse at points $A$ and $B$, and the line $l_{2}: y=kx+m_{2} (m_{1} \neq m_{2})$ intersects the ellipse at points $C$ and $D$, and $|AB|=|CD|$.
$(i)$ Find the value of $m_{1}+m_{2}$;
$(ii)$ Find the maximum value of the area $S$ of the quadrilateral $ABCD$.

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