
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

For a plane set \( S \), define \( d(S) \), the diameter of \( S \), to be \( \sup \{PQ : P, Q \in S\} \). Let \( K \) be a triangle with sides 3, 4, 5 and its interior. If \( K = H_1 \cup H_2 \cup H_3 \cup H_4 \), what is the smallest possible value of \( \max(d(H_1), d(H_2), d(H_3), d(H_4)) \)?

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