
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 large triangular array of squares has one square in the first row, two in the second, and generally, $k$ squares in the $k$th row for $1 \leq k \leq 14$. With the exception of the bottom row, each square rests on two squares in the row immediately below. Each square of the fourteenth row is labeled with 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 immediately below it. Determine the number of ways in which the squares in the fourteenth row can be filled so that the number in the top square (first row) is a multiple of $5$.

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