
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
Adam and Bettie are playing a game. They take turns generating a random number between  $0$  and  $127$  inclusive. The numbers they generate are scored as follows: $\bullet$  If the number is zero, it receives no points. $\bullet$  If the number is odd, it receives one more point than the number one less than it. $\bullet$  If the number is even, it receives the same score as the number with half its value.
if Adam and Bettie both generate one number, the probability that they receive the same score is  $\frac{p}{q}$  for relatively prime positive integers  $p$  and  $q$ . Find  $p$ .

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