
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 an integer  $n\geq 3$ . For each  $3\times3$  squares on the grid, call this  $3\times3$  square isolated if the center unit square is white and other 8 squares are black, or the center unit square is black and other 8 squares are white. 

Now suppose one can paint an infinite grid by white or black, so that one can select an  $a\times b$  rectangle which contains at least  $n^2-n$  isolated  $3\times 3$  square. Find the minimum of  $a+b$  that such thing can happen.

(Note that  $a,b$  are positive reals, and selected  $a\times b$  rectangle may have sides not parallel to grid line of the infinite grid.)

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