
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
Chords on a sphere are segments connecting two points on the sphere's surface. For a sphere with a radius of 4, the lengths of two chords AB and CD are, respectively, $2\sqrt{7}$ and $4\sqrt{3}$. M and N are the midpoints of AB and CD, respectively. The endpoints of both chords move along the sphere's surface. Consider the following four propositions:  
① Chords AB and CD may intersect at point M;  
② Chords AB and CD may intersect at point N;  
③ The maximum value of MN is 5;  
④ The minimum value of MN is 1.  
The correct statement(s) is/are indicated by the number(s) \_\_\_\_\_\_ .

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