
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
**p1.** The Evergreen School booked buses for a field trip. Altogether,  $138$  people went to West Lake, while  $115$  people went to East Lake. The buses all had the same number of seats and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were on each bus?**p2.** In New Scotland there are three kinds of coins:  $1$  cent,  $6$  cent, and  $36$  cent coins. Josh has  $99$  of the  $36$ -cent coins (and no other coins). He is allowed to exchange a  $36$  cent coin for  $6$  coins of  $6$  cents, and to exchange a  $6$  cent coin for  $6$  coins of  $1$  cent. Is it possible that after several exchanges Josh will have  $500$  coins?**p3.** Find all solutions  $a, b, c, d, e, f, g, h$  if these letters represent distinct digits and the following multiplication is correct: $\begin{tabular}{ccccc}
 &  & a & b & c 
+ &  &  & d & e 
\hline
 & f & a & g & c 
x & b & b & h &  
\hline
f & f & e & g & c 
\end{tabular}$ **p4.** Is it possible to find a rectangle of perimeter  $10$  m and cut it in rectangles (as many as you want) so that the sum of the perimeters is  $500$  m?**p5.** The picture shows a maze with chambers (shown as circles) and passageways (shown as segments). A cat located in chamber  $C$  tries to catch a mouse that was originally in the chamber  $M$ . The cat makes the first move, moving from chamber  $C$  to one of the neighboring chambers. Then the mouse moves, then the cat, and so forth. At each step, the cat and the mouse can move to any neighboring chamber or not move at all. The cat catches the mouse by moving into the chamber currently occupied by the mouse. Can the cat get the mouse?
![Image](https://cdn.artofproblemsolving.com/attachments/9/9/25f61e1499ff1cfeea591cb436d33eb2cdd682.png)



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