
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
Given a regular tetrahedron \(ABCD\) with edge length 2, there is a point \(P\) on edge \(AB\) such that \(AP < 1\). A cut is made through point \(P\) perpendicular to the edge \(AB\) and continues through the faces but stops at a certain point. When the cut stops, the length of the cut on face \(ABD\) is \(PM = 1\) and the length of the cut on face \(ABC\) is \(PN = \frac{2}{3}\). Find the length \(MN\).

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