
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.** A bus leaves San Mateo with  $n$  fairies on board. When it stops in San Francisco, each fairy gets off, but for each fairy that gets off,  $n$  fairies get on. Next it stops in Oakland where  $6$  times as many fairies get off as there were in San Mateo. Finally the bus arrives at Berkeley, where the remaining  $391$  fairies get off. How many fairies were on the bus in San Mateo?**p2.** Let  $a$  and  $b$  be two real solutions to the equation  $x^2 + 8x - 209 = 0$ . Find  $\frac{ab}{a+b}$  . Express your answer as a decimal or a fraction in lowest terms.**p3.** Let  $a$ ,  $b$ , and  $c$  be positive integers such that the least common multiple of  $a$  and  $b$  is  $25$  and the least common multiple of  $b$  and  $c$  is  $27$ . Find  $abc$ .**p4.** It takes Justin  $15$  minutes to finish the Speed Test alone, and it takes James  $30$  minutes to finish the Speed Test alone. If Justin works alone on the Speed Test for  $3$  minutes, then how many minutes will it take Justin and James to finish the rest of the test working together? Assume each problem on the Speed Test takes the same amount of time.**p5.** Angela has  $128$  coins.  $127$  of them have the same weight, but the one remaining coin is heavier than the others. Angela has a balance that she can use to compare the weight of two collections of coins against each other (that is, the balance will not tell Angela the weight of a collection of coins, but it will say which of two collections is heavier). What is the minumum number of weighings Angela must perform to guarantee she can determine which coin is heavier?



PS. You had better use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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