
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 $A(-1,0)$, $B(1,0)$, and $M$, $N$ are two moving points on the line $x=a(a > 1)$, and the line $AM$ is perpendicular to the line $BN$. The curve $C$ is a circle with diameter $MN$.

$(1)$ Determine the positional relationship between the origin $O$ and the circle $C$.

$(2)$ When $M$, $N$ are symmetric about the $x$-axis, find the standard equation of the circle $C$.

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