
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
Consider a checkerboard of $14$ rows and $15$ columns with numbers written in each square starting from the upper left corner so that the first row is numbered $1, 2, \ldots, 15.$ The second row continues from $16, 17, \ldots, 30$, and so forth downwards. The board is then renumbered such that the left column, from top to bottom, is $1, 2, \ldots, 14$, the second column $15, 16, \ldots, 28$, and continues to the last column. Determine the sum of the numbers in squares where these two numbering systems produce the same number.

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