
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 real number between 0 and 1. Jocelin has a coin that lands heads with probability \( p \) and tails with probability \( 1-p \); she also has a number written on a blackboard. Each minute, she flips the coin, and if it lands heads, she replaces the number \( x \) on the blackboard with \( 3x + 1 \); if it lands tails, she replaces it with \( \frac{x}{2} \). Given that there are constants \( a, b \) such that the expected value of the value written on the blackboard after \( t \) minutes can be written as \( at + b \) for all positive integers \( t \), compute \( p \).

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