
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 $1 \leq k \leq 12$. Each square except those in the bottom row rests on two squares in the row immediately below. Each square in the twelfth row (bottommost) contains either a $0$ or a $1$. Numbers are then placed into the other squares, with each square's entry being the sum of the entries in the two squares it rests on. How many initial distributions of $0$s and $1$s in the bottom row result in the number in the top square being a multiple of $4$?

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