
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
The segment connecting two points on the surface of a sphere is called a chord of the sphere. 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. Both ends of each chord move on the surface of the sphere, and there are the following four propositions:  
① The maximum value of MN is 5.  
② The chords AB and CD may intersect at point M.  
③ The minimum value of MN is 1.  
④ The chords AB and CD may intersect at point N.  
Among these, the true propositions are \_\_\_\_\_\_.

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