
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 vertices of a $3 \times 1 \times 1$ rectangular prism are $A, B, C, D, E, F, G$, and $H$ so that $A E, B F$, $C G$, and $D H$ are edges of length 3. Point $I$ and point $J$ are on $A E$ so that $A I=I J=J E=1$. Similarly, points $K$ and $L$ are on $B F$ so that $B K=K L=L F=1$, points $M$ and $N$ are on $C G$ so that $C M=M N=N G=1$, and points $O$ and $P$ are on $D H$ so that $D O=O P=P H=1$. For every pair of the 16 points $A$ through $P$, Maria computes the distance between them and lists the 120 distances. How many of these 120 distances are equal to $\sqrt{2}$?

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