
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 $i$ as the imaginary unit, $a$, $b \in \mathbb{R}$, let complex numbers $z_{1} = 2a - \sqrt{3}i$, $z_{2} = 2b + i$, $z_{3} = a + bi$, and $|z_{3}| = 1$.
$(1)$ If $z_{1} - z_{2}$ is a purely imaginary number, find $z_{3}$.
$(2)$ If the points corresponding to complex numbers $z_{1}$ and $z_{2}$ on the complex plane are $A$ and $B$ respectively, with $O$ as the origin of the complex plane.
① Does there exist real numbers $a$, $b$ such that the vector $\overrightarrow{OB}$, after counterclockwise rotation of $90^{\circ}$, coincides with the vector $\overrightarrow{OA}$? If yes, find the values of $a$ and $b$; if not, explain why.
② If $O$, $A$, $B$ are non-collinear points, and the area of $\triangle ABO$ is denoted as $S(a,b)$, find $S(a,b)$ and its maximum value.

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