
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
With about six hours left on the van ride home from vacation, Wendy looks for something to do.  She starts working on a project for the math team.

There are sixteen students, including Wendy, who are about to be sophomores on the math team.  Elected as a math team officer, one of Wendy's jobs is to schedule groups of the sophomores to tutor geometry students after school on Tuesdays.  The way things have been done in the past, the same number of sophomores tutor every week, but the same group of students never works together.  Wendy notices that there are even numbers of groups she could select whether she chooses  $4$  or  $5$  students at a time to tutor geometry each week:
\begin{align*}\dbinom{16}4&=1820,\dbinom{16}5&=4368.\end{align*} 
Playing around a bit more, Wendy realizes that unless she chooses all or none of the students on the math team to tutor each week that the number of possible combinations of the sophomore math teamers is always even.  This gives her an idea for a problem for the  $2008$  Jupiter Falls High School Math Meet team test:

\[\text{How many of the 2009 numbers on Row 2008 of Pascal's Triangle are even?}\]

Wendy works the solution out correctly.  What is her answer?

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