
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
On a number line, points with integer coordinates are colored red and blue according to the following rules:
a) Points whose coordinate difference is 7 must be the same color.
b) Points with coordinates 20 and 14 must be colored red, and points with coordinates 71 and 143 must be colored blue.

In how many ways can all points with integer coordinates be colored while obeying these rules?

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