
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 an Olympic camp, there are \( a \) trainees and \( b \) examiners, where \( b \) is odd and at least 3. Each examiner decides for each trainee whether they have passed or failed the camp. Let \( k \) be an integer such that no two examiners can have the same opinion on more than \( k \) trainees. Show the inequality

$$
\frac{k}{a} \geqslant \frac{b-1}{2b}
$$

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