
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
In each cell of an $8 \times 8$ square, one of the diagonals is drawn. Consider the union of these 64 diagonals. It consists of several connected components (a point belongs to one component if it can be reached from another point by one or more diagonals). Can the number of these components be more than
a) 15;

b) 20?

c) In a similar problem with an $n \times n$ square grid, can the number of components be more than $n^2 / 4$ (for $n > 8$)?

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