
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
[asy]
draw((0,1)--(4,1)--(4,2)--(0,2)--cycle);
draw((2,0)--(3,0)--(3,3)--(2,3)--cycle);
draw((1,1)--(1,2));
label("1",(0.5,1.5));
label("2",(1.5,1.5));
label("32",(2.5,1.5));
label("16",(3.5,1.5));
label("8",(2.5,0.5));
label("6",(2.5,2.5));
[/asy]
The image above is a net of a unit cube. Let  $n$  be a positive integer, and let  $2n$  such cubes are placed to build a  $1 \times 2 \times n$  cuboid which is placed on a floor. Let  $S$  be the sum of all numbers on the block visible (not facing the floor). Find the minimum value of  $n$  such that there exists such cuboid and its placement on the floor so  $S > 2011$ .

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