
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
Cities $P$, $Q$, $R$, $S$, $T$, and $U$ are connected by roads $\widetilde{PQ}$, $\widetilde{PS}$, $\widetilde{PT}$, $\widetilde{PU}$, $\widetilde{QR}$, $\widetilde{QS}$, $\widetilde{RS}$, $\widetilde{RT}$, $\widetilde{SU}$, $\widetilde{UT}$. How many different routes are there from $P$ to $Q$ that use each road exactly once, assuming each city may need to be visited more than once?

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