
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 \( X = \{1, 2, \ldots, 98\} \). Call a subset of \( X \) good if it satisfies the following conditions:
1. It has 10 elements.
2. If it is partitioned in any way into two subsets of 5 elements each, then one subset has an element coprime to each of the other 4, and the other subset has an element which is not coprime to any of the other 4.

Find the smallest \( n \) such that any subset of \( X \) of \( n \) elements has a good subset.

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