
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
How many positive integers \( 2 \leq a \leq 101 \) have the property that there exists a positive integer \( N \) for which the last two digits in the decimal representation of \( a^{2^{n}} \) are the same for all \( n \geq N \)?

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