
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.** Let  $A\% B = BA - B - A + 1$ . How many digits are in the number  $1\%(3\%(3\%7))$  ?**p2.**Three circles, of radii  $1, 2$ , and  $3$  are all externally tangent to each other. A fourth circle is drawn which passes through the centers of those three circles. What is the radius of this larger circle?**p3.** Express  $\frac13$  in base  $2$  as a binary number. (Which, similar to how demical numbers have a decimal point, has a “binary point”.)**p4.** Isosceles trapezoid  $ABCD$  with  $AB$  parallel to  $CD$  is constructed such that  $DB = DC$ . If  $AD = 20$ ,  $AB = 14$ , and  $P$  is the point on  $AD$  such that  $BP + CP$  is minimized, what is  $AP/DP$ ?**p5.** Let  $f(x) = \frac{5x-6}{x-2}$  . Define an infinite sequence of numbers  $a_0, a_1, a_2,....$  such that  $a_{i+1} = f(a_i)$  and  $a_i$  is always an integer. What are all the possible values for  $a_{2014}$  ?**p6.**  $MATH$  and  $TEAM$  are two parallelograms. If the lengths of  $MH$  and  $AE$  are  $13$  and  $15$ , and distance from  $AM$  to  $T$  is  $12$ , find the perimeter of  $AMHE$ .**p7.** How many integers less than  $1000$  are there such that  $n^n + n$  is divisible by  $5$  ?**p8.**  $10$  coins with probabilities of  $1, 1/2, 1/3 ,..., 1/10$  of coming up heads are flipped. What is the probability that an odd number of them come up heads?**p9.** An infinite number of coins with probabilities of  $1/4, 1/9, 1/16, ...$  of coming up heads are all flipped. What is the probability that exactly  $ 1$  of them comes up heads?**p10.** Quadrilateral  $ABCD$  has side lengths  $AB = 10$ ,  $BC = 11$ , and  $CD = 13$ . Circles  $O_1$  and  $O_2$  are inscribed in triangles  $ABD$  and  $BDC$ . If they are both tangent to  $BD$  at the same point  $E$ , what is the length of  $DA$  ?


PS. You had better use hide for answers.

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