
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 the right parallelopiped  $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$ , with  $AB=12\sqrt{3}$  cm and  $AA^{\prime}=18$  cm, we consider the points  $P\in AA^{\prime}$  and  $N\in A^{\prime}B^{\prime}$  such that  $A^{\prime}N=3B^{\prime}N$ . Determine the length of the line segment  $AP$  such that for any position of the point  $M\in BC$ , the triangle  $MNP$  is right angled at  $N$ .

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