
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
Let \( X \) be the number of sequences of integers \( a_1, a_2, \ldots, a_{2047} \) that satisfy all of the following properties:
- Each \( a_i \) is either 0 or a power of 2.
- \( a_i = a_{2i} + a_{2i+1} \) for \( 1 \leq i \leq 1023 \).
- \( a_1 = 1024 \).

Find the remainder when \( X \) is divided by 100.

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