
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 $1 \leq k \leq 13.$ Like before, each square rests on two squares from the row immediately below. In each square of the thirteenth row, a $0$ or a $1$ is placed. Numbers are then placed into the remaining squares such that the number in each square is the sum of the numbers in the two squares directly below it. Determine the number of ways to arrange $0$'s and $1$'s in the thirteenth row so that the number in the top square is a multiple of $6$.

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