
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
<u>Set 1</u>**1.1** Compute the number of real numbers x such that the sequence  $x$ ,  $x^2$ ,  $x^3$ , $ x^4$ ,  $x^5$ ,  $...$  eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence  $1$ ,  $2$ ,  $3$ ,  $4$ ,  $5$ ,  $6$ ,  $5$ ,  $6$ ,  $5$ ,  $6$ ,  $...$  is eventually repeating with repeating block  $5$ ,  $6$ .)**1.2** Let  $T$  be the answer to the previous problem. Nicole has a broken calculator which, when told to multiply  $a$  by  $b$ , starts by multiplying  $a$  by  $b$ , but then multiplies that product by b again, and then adds  $b$  to the result. Nicole inputs the computation “ $k \times k$ ” into the calculator for some real number  $k$  and gets an answer of  $10T$ . If she instead used a working calculator, what answer should she have gotten?**1.3** Let  $T$  be the answer to the previous problem. Find the positive difference between the largest and smallest perfect squares that can be written as  $x^2 + y^2$  for integers  $x, y$  satisfying  $\sqrt{T} \le x \le T$  and  $\sqrt{T} \le y \le T$ .


PS. You should use hide for answers. 

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