
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
In the rectangular coordinate system on a plane, there are two points $A(-1,3 \sqrt{3})$ and $B(1, \sqrt{3})$. Draw a circle with the origin as the center and $r(r > 0)$ as the radius. The circle intersects with the ray $y=- \sqrt{3}x (x < 0)$ at point $M$ and intersects with the positive half of the $x$-axis at point $N$. Find the minimum value of $|AM|+|BN|$ as $r$ varies.

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