
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 \( p \) be a positive integer. Carlos and Alice play a game on a whiteboard where they interact with numbers derived from powers of integers. Carlos starts by writing the smallest power of 2 (exponentiation, not square) with \( p+1 \) digits. Every time Carlos writes a number, Alice erases the last \( p \) digits of it. Carlos then writes the next higher power of 2, Alice erases the last \( p \) digits, and this process continues until the last two numbers that remain on the board differ by at least 3. Let \( h(p) \) be the smallest positive integer not written on the board. Determine \( h(1) \).

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