
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 the first 30 rows of Pascal's triangle, beginning with row zero. A row is said to be "pure" if, except for the $1$ at each end, all numbers in the row are either all even or all odd. For example, Row 4 consists only of the numbers 1, 4, 6, 4, 1, where all middle numbers are even, thus making it a "pure" row. Count how many of the first 30 rows of Pascal's triangle are "pure" rows.

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