
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
Given \( N \) points on a plane, the distances between every pair of points \( M_i \) and \( M_j \) (where \( i \) and \( j \) are any integers from 1 to \( N \)) need to be equal to predefined values \( r_{ij} \).

a) Is it possible to construct such a set of \( N \) points if it is known that any subset of 5 out of these \( N \) points can be constructed with the given distances?

b) Is it sufficient to require that any subset of 4 out of these \( N \) points can be constructed with the given distances?

c) How does the problem change if the points are constructed in 3-dimensional space instead of a plane? What is the smallest number \( k \) such that the constructibility of any \( k \) out of the \( N \) points guarantees the constructibility of all \( N \) points?

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