
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 a school, there are  $n$  students and some of them are friends each other. (Friendship is mutual.) Define  $ a, b $  the minimum value which satisfies the following conditions:
(1) We can divide students into  $ a $  teams such that two students in the same team are always friends.
(2) We can divide students into  $ b $  teams such that two students in the same team are never friends.
Find the maximum value of  $ N = a+b $  in terms of  $n$ .

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