
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 \( 1 \leq k \leq 13 \). Except for the bottom row, each square is based on the sum of the entries of two squares directly beneath it. Initially, each square in the thirteenth row contains either a \( 0 \) or a \( 1 \). Calculate the number of ways to fill the thirteenth row such that the value of the square at the top of the triangle is divisible by 5.

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