
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
Place the numbers $1-8$ on the eight vertices of a cube, then at the midpoint of each edge write the average of the two numbers at the endpoints of that edge. If the numbers at the four midpoints of the top face and the four midpoints of the bottom face are all integers, how many of the numbers at the other four midpoints are not integers?

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