
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
You are given an odd number  $n\ge 3$ . For every pair of integers  $(i, j)$  with  $1\le i \le j \le n$  there is a domino, with  $i$  written on one its end and with  $j$  written on another (there are  $\frac{n(n+1)}{2}$  domino overall). Amin took this dominos and started to put them in a row so that numbers on the adjacent sides of the dominos are equal. He has put  $k$  dominos in this way, got bored and went away. After this Anton came to see this  $k$  dominos, and he realized that he can't put all the remaining dominos in this row by the rules. For which smallest value of  $k$  is this possible? 

*Oleksii Masalitin*

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