
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 blue square of side length  $10$  is laid on top of a coordinate grid with corners at  $(0,0)$ ,  $(0,10)$ ,  $(10,0)$ , and  $(10,10)$ . Red squares of side length  $2$  are randomly placed on top of the grid, changing the color of a  $2\times2$  square section red. Each red square when placed lies completely within the blue square, and each square's four corners take on integral coordinates. In addition, randomly placed red squares may overlap, keeping overlapped regions red. Compute the expected value of the number of red squares necessary to turn the entire blue square red, rounded to the nearest integer. Your score will be given by  $\lfloor25\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor$ , where  $A$  is your answer and  $C$  is the actual answer.

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