
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
Let  $n$  be a fixed positive integer. Oriol has  $n$  cards, each of them with a  $0$  written on one side and  $1$  on the other. We place these cards in line, some face up and some face down (possibly all on the same side). We begin the following process consisting of  $n$  steps:

1) At the first step, Oriol flips the first card
2) At the second step, Oriol flips the first card and second card
.
.
.
n) At  the last step Oriol flips all the cards

Let  $s_0, s_1, s_2, \dots, s_n$  be the sum of the numbers seen in the cards at the beggining, after the first step, after the second step,  $\dots$  after the last step, respectively.

          a) Find the greatest integer  $k$  such that, no matter the initial card configuration, there exists at least  $k$  distinct numbers between  $s_0, s_1, \dots, s_n$ .
          b) Find all positive integers  $m$  such that, for each initial card configuration, there exists an index  $r$  such that  $s_r = m$ .

*Proposed by Dorlir Ahmeti*

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