
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 \( 1 \leq k \leq 9 \). Each square, except those in the bottom row, contains a number that is the sum of the numbers in the two squares directly below it. In each square of the ninth row, a 0 or a 1 is placed. Determine for how many initial distributions of 0's and 1's in the bottom row, the number in the top square is a multiple of 2.

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