
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
Write the numbers \( x_k = \lfloor k \sqrt{2} \rfloor \) (for \( k = 1, 2, \ldots \)) in a row, and below them the integers \( 0 < y_1 < y_2 < \ldots \) that do not appear in the \( x_k \) sequence. Determine the difference \( y_k - x_k \) as a function of \( k \).

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