
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
Line \(L_1\) is defined by \( \{ (t + 1, 2t - 4, -3t + 5) : t \text{ real} \} \) and line \(L_2\) is defined by \( \{ (4t - 12, -t + 8, t + 17) : t \text{ real} \} \). Find the smallest sphere that touches both \(L_1\) and \(L_2\).

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