
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 $\Gamma _{1}$: $\frac{{x}^{2}}{4}-\frac{{y}^{2}}{{b}^{2}}=1$ and the circle $\Gamma _{2}$: $x^{2}+y^{2}=4+b^{2}(b \gt 0)$ intersect at point $A(x_{A}$,$y_{A})$ (in the first quadrant). The curve $\Gamma$ is the part of $\Gamma _{1}$ and $\Gamma _{2}$ that satisfies $|x| \gt x_{A}$.  
$(1)$ If $x_{A}=\sqrt{6}$, find the value of $b$;  
$(2)$ When $b=\sqrt{5}$, the intersection points of $\Gamma _{2}$ with the $x$-axis are denoted as $F_{1}$ and $F_{2}$. $P$ is a point on the curve $\Gamma$, in the first quadrant, and $|PF_{1}|=8$. Find $\angle F_{1}PF_{2}$;  
$(3)$ A line $l$ passing through point $D(0$,$\frac{{b}^{2}}{2}+2)$ with a slope of $-\frac{b}{2}$ intersects the curve $\Gamma$ at only two points, denoted as $M$ and $N$. Express $\overrightarrow{OM}\cdot \overrightarrow{ON}$ in terms of $b$, and find the range of values for $\overrightarrow{OM}\cdot \overrightarrow{ON}$.

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