
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
Define a \emph{crossword puzzle} to be a  $15 \times 15$  grid of squares, each of which is either black or white. In a crossword puzzle, define a \emph{word} to be a sequence of one or more consecutive white squares in a row or column such that the squares immediately before and after the sequence both are either black or nonexistent. (The latter case would occur if an end of a word coincides with an end of a row or column of the grid.)  A crossword puzzle is \emph{tasty} if every word consists of an even number of white squares. Compute the sum of all nonnegative integers  $n$  such that there exists a tasty crossword puzzle with exactly  $n$  white squares.

*Proposed by Luke Robitaille*

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