
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
1. Rowan is given a square grid whose side length is an odd integer. The square is divided into 1 by 1 squares. Rowan shades each 1 by 1 square along the two diagonals and does not shade any of the remaining squares.

(a) Rowan is given a square grid that is 7 by 7. How many 1 by 1 squares does he shade?

(b) Rowan is given a square grid that is 101 by 101. Explain why the number of 1 by 1 squares that he shades is 201.

(c) Rowan is given another square grid with odd side length, and the number of 1 by 1 squares that he shades is 41. How many unshaded 1 by 1 squares are there in the grid?

(d) Rowan is given a fourth square grid with odd side length. After shading the squares on the diagonals, there are 196 unshaded 1 by 1 squares. How many 1 by 1 squares does this grid contain in total?

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