
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 points $A(-2,0)$, $B(2,0)$, and a moving point $M$ such that $\angle AMB = 2\theta$, and $|\overrightarrow{AM}|\cdot |\overrightarrow{BM}|=\frac{4}{{\cos }^{2}\theta}$.

$(1)$ Find the value of $|\overrightarrow{AM}|+|\overrightarrow{BM}|$ and write down the equation of the trajectory curve $C$ of point $M$;

$(2)$ A moving line $l: y=kx+m$ intersects curve $C$ at points $P$ and $Q$, and $OP \perpendicular OQ$. Determine whether there exists a circle $x^{2}+y^{2}=r^{2}$ such that $l$ is exactly the tangent line of the circle. If it exists, find $r$; if not, explain why.

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