
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 magician has  $300$  cards with numbers from  $1$  to  $300$  written on them, each number on exactly one card. The magician then lays these cards on a  $3 \times 100$  rectangle in the following way - one card in each unit square so that the number cannot be seen and cards with consecutive numbers are in neighbouring squares. Afterwards, the magician turns over  $k$  cards of his choice. What is the smallest value of  $k$  for which it can happen that the opened cards definitely determine the exact positions of all other cards?

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