
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
A sequence  $x_1, x_2, ..., x_n, ...$  consists of an initial block of  $p$  positive distinct integers that then repeat periodically. This means that  $\{x_1, x_2, \dots, x_p\}$  are  $p$  distinct positive integers and  $x_{n+p}=x_n$  for every positive integer  $n$ . The terms of the sequence are not known and the goal is to find the period  $p$ . To do this, at each move it possible to reveal the value of a term of the sequence at your choice.

(a) Knowing that  $1 \le p \le 10$ , find the least  $n$  such that there is a strategy which allows to find  $p$  revealing at most  $n$  terms of the sequence.
(b) Knowing that  $p$  is one of the first  $k$  prime numbers, find for which values of  $k$  there exist a strategy that allows to find  $p$  revealing at most  $5$  terms of the sequence.

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