
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
Imagine a triangular array of squares, where there is one square in the first row, two in the second, continuing up to 10 squares in the tenth row. Except for the bottom row, each square rests on two squares immediately below it. We start by placing either a $0$ or $2$ in each square of the tenth row. The number in each subsequent square is the sum of the entries in the two squares directly below it. Determine how many initial distributions of $0$'s and $2$'s in the tenth row will make the number in the top square a multiple of $4$.

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