
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
**p4.** What is gcd  $(2^6 - 1, 2^9 - 1)$ ?**p5.** Sarah is walking along a sidewalk at a leisurely speed of  $\frac12$  m/s. Annie is some distance behind her, walking in the same direction at a faster speed of  $s$  m/s. What is the minimum value of  $s$  such that Sarah and Annie spend no more than one second within one meter of each other?**p6.** You have a choice to play one of two games. In both games, a coin is flipped four times. In game  $1$ , if (at least) two flips land heads, you win. In game  $2$ , if (at least) two consecutive flips land heads, you win. Let  $N$  be the number of the game that gives you a better chance of winning, and let  $p$  be the absolute difference in the probabilities of winning each game. Find  $N + p$ .


PS. You should use hide for answers.

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