
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
A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 11$. Each square in each row (except for the last row) is positioned directly above two squares of the row immediately below it. In each square of the eleventh row, a number which is either $0$ or $1$ is placed. Numbers are placed into the other squares in such a way that each square's number is the sum of the numbers in the two squares directly below it. How many ways can the $0$s and $1$s be arranged in the eleventh row such that the number in the top square is a multiple of $5$?

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