
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
There are three types of piece shown as below. Today Alice wants to cover a  $100 \times 101$  board with these pieces without gaps and overlaps. Determine the minimum number of  $1\times 1$  pieces should be used to cover the whole board and not exceed the board. (There are an infinite number of these three types of pieces.)
[asy]
size(9cm,0);
defaultpen(fontsize(12pt));
draw((9,10) -- (59,10) -- (59,60) -- (9,60) -- cycle);
draw((59,10) -- (109,10) -- (109,60) -- (59,60) -- cycle);
draw((9,60) -- (59,60) -- (59,110) -- (9,110) -- cycle);
draw((9,110) -- (59,110) -- (59,160) -- (9,160) -- cycle);
draw((109,10) -- (159,10) -- (159,60) -- (109,60) -- cycle);
draw((180,11) -- (230,11) -- (230,61) -- (180,61) -- cycle);
draw((180,61) -- (230,61) -- (230,111) -- (180,111) -- cycle);
draw((230,11) -- (280,11) -- (280,61) -- (230,61) -- cycle);
draw((230,61) -- (280,61) -- (280,111) -- (230,111) -- cycle);
draw((280,11) -- (330,11) -- (330,61) -- (280,61) -- cycle);
draw((280,61) -- (330,61) -- (330,111) -- (280,111) -- cycle);
draw((330,11) -- (380,11) -- (380,61) -- (330,61) -- cycle);
draw((330,61) -- (380,61) -- (380,111) -- (330,111) -- cycle);
draw((401,11) -- (451,11) -- (451,61) -- (401,61) -- cycle);
[/asy]
*Proposed by amano_hina*

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