
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
a) Find all possible arrangements of four points on the plane, where the pairwise distances between these points take only two distinct values \(a\) and \(b\). List all possible values of the ratio \(\frac{b}{a}\) for which such arrangements are possible.

b) Find all possible arrangements of \(n\) points on the plane such that all pairwise distances between these points take one of two values \(a\) or \(b\). For which values of \(n\) do such arrangements exist?

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