
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
Let $T$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le3,$ $0\le y\le4,$ and $z \in \{0,1,3,4\}.$ Two distinct points are randomly chosen from $T.$ The probability that the midpoint of the segment they determine also belongs to $T$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p + q.$

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