
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
Let's call a number \( \mathrm{X} \) "20-supporting" if for any 20 real numbers \( a_{1}, \ldots, a_{20} \), whose sum is an integer, there is at least one number for which \( \left|a_{i}-\frac{1}{2}\right| \geq X \).

Find the largest 20-supporting \( X \), rounded to the nearest thousandth by standard mathematical rules.

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