
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
One night, over dinner Jerry poses a challenge to his younger children: "Suppose we travel  $50$  miles per hour while heading to our final vacation destination..."

Hannah teases her husband, "You  $\textit{would}$  drive that  $\textit{slowly}\text{!}$ "

Jerry smirks at Hannah, then starts over, "So that we get a good view of all the beautiful landscape your mother likes to photograph from the passenger's seat, we travel at a constant rate of  $50$  miles per hour on the way to the beach.  However, on the way back we travel at a faster constant rate along the exact same route.  If our faster return rate is an integer number of miles per hour, and our average speed for the  $\textit{whole round trip}$  is  $\textit{also}$  an integer number of miles per hour, what must be our speed during the return trip?"

Michael pipes up, "How about  $4950$  miles per hour?!"

Wendy smiles, "For the sake of your  $\textit{other}$  children, please don't let  $\textit{Michael}$  drive."

Jerry adds, "How about we assume that we never  $\textit{ever}$  drive more than  $100$  miles per hour.  Michael and Wendy, let Josh and Alexis try this one."

Joshua ignores the problem in favor of the huge pile of mashed potatoes on his plate.  But Alexis scribbles some work on her napkin and declares the correct answer.  What answer did Alexis find?

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