
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 $E$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ with eccentricity $e=\frac{\sqrt{3}}{2}$, and passing through the fixed point $P(\sqrt{3}$,$\frac{1}{2})$. <br/>$(Ⅰ)$ Find the equation of the ellipse $E$; <br/>$(Ⅱ)$ Determine if there exists a line $y=-x+m$ that intersects the ellipse at points $A$ and $B$, satisfying $OA\bot OB$. If it exists, find the value of $m$; if not, explain the reason.

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