
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
Let  $n>1$  be an even positive integer. An  $2n \times 2n$  grid of unit squares is given, and it is partitioned into  $n^2$  contiguous  $2 \times 2$  blocks of unit squares. A subset  $S$  of the unit squares satisfies the following properties:

(i) For any pair of squares  $A,B$  in  $S$ , there is a sequence of squares in  $S$  that starts with  $A$ , ends with  $B$ , and has any two consecutive elements sharing a side; and 

(ii) In each of the  $2 \times 2$  blocks of squares, at least one of the four squares is in  $S$ . 

An example for  $n=2$  is shown below, with the squares of  $S$  shaded and the four  $2 \times 2$  blocks of squares outlined in bold. 

[asy]
size(2.5cm);
fill((0,0)--(4,0)--(4,1)--(0,1)--cycle,mediumgrey);
fill((0,0)--(0,4)--(1,4)--(1,0)--cycle,mediumgrey);
fill((0,3)--(4,3)--(4,4)--(0,4)--cycle,mediumgrey);
fill((3,0)--(3,4)--(4,4)--(4,0)--cycle,mediumgrey);
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
draw((1,0)--(1,4));
draw((2,0)--(2,4),linewidth(1));
draw((3,0)--(3,4));
draw((0,1)--(4,1));
draw((0,2)--(4,2),linewidth(1));
draw((0,3)--(4,3));
[/asy]

In terms of  $n$ , what is the minimum possible number of elements in  $S$ ?

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