
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  $n \geq 2$  be a natural number, and let  $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$  be a permutation of  $\left(1;\;2;\;...;\;n\right)$ . For any integer  $k$  with  $1 \leq k \leq n$ , we place  $a_k$  raisins on the position  $k$  of the real number axis. [The real number axis is the  $x$ -axis of a Cartesian coordinate system.]

Now, we place three children A, B, C on the positions  $x_A$ ,  $x_B$ ,  $x_C$ , each of the numbers  $x_A$ ,  $x_B$ ,  $x_C$  being an element of  $\left\{1;\;2;\;...;\;n\right\}$ . [It is not forbidden to place different children on the same place!]

For any  $k$ , the  $a_k$  raisins placed on the position  $k$  are equally handed out to those children whose positions are next to  $k$ . [So, if there is only one child lying next to  $k$ , then he gets the raisin. If there are two children lying next to  $k$  (either both on the same position or symmetric with respect to  $k$ ), then each of them gets one half of the raisin. Etc..]

After all raisins are distributed, a child is unhappy if he could have received more raisins than he actually has received if he had moved to another place (while the other children would rest on their places).

For which  $n$  does there exist a configuration  $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$  and numbers  $x_A$ ,  $x_B$ ,  $x_C$ , such that all three children are happy?

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