
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 hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with its asymptote equations $y = \pm \frac{\sqrt{3}}{3}x$, and the left focus being $F$, through points $A(a,0)$ and $B(0,-b)$ line $l$ is drawn, and the distance from the origin to line $l$ is $\frac{\sqrt{3}}{2}$.
$(1)$ Find the equation of the hyperbola;
$(2)$ Suppose a line $y = x + m$ intersects the hyperbola at two distinct points $C$ and $D$. Is there a real number $m$ such that the circle with diameter $CD$ passes through the left focus $F$ of the hyperbola? If it exists, find the value(s) of $m$; if not, explain why.

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