
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) How many squares make up the board with 11 rows and 11 columns?

(b) The diagonal whose squares are shaded separates the board into two regions: one above and one below. How many squares make up each region? Is it possible to calculate this number without counting square by square?

(c) With the help of the board, is it possible to calculate the sum $1+2+\cdots+10$? Explain how.

(d) With the help of another board, using reasoning similar to the previous item, is it possible to calculate the sum $1+2+\cdots+100$? What should be the number of rows and columns of the board? What is the value of the sum?

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