
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
A line segment connecting two points on the surface of a sphere is called a chord of the sphere. In a sphere with a radius of 4, the lengths of two chords $AB$ and $CD$ are $2 \sqrt{7}$ and $4 \sqrt{3}$ respectively. $M$ and $N$ are the midpoints of $AB$ and $CD$ respectively, with each end point of the two chords moving on the surface of the sphere. The following four statements are given:
(1) The chords $AB$ and $CD$ may intersect at point $M$.
(2) The 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.
Determine the true statement(s) among the above.

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