
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 tetrahedron \(ABCDA\), faces \(ABC\) and \(BCD\) are both equilateral triangles with side length 2. Additionally, \(AD = 2\sqrt{2}\). Points \(M\) and \(N\) are the midpoints of edges \(AB\) and \(CD\), respectively. Calculate the shortest distance between \(M\) and \(N\) in the tetrahedron.

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