
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
In the set of integers $1, 2, \cdots, 99$, determine the smallest value of $k$ such that among any $k$ chosen numbers, there are always two distinct numbers $a$ and $b$ ($a  \neq b$) that satisfy $\frac{1}{2} \leqslant \frac{b}{a} \leqslant 2$.

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