
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
Find the equation of the smallest sphere which is tangent to both of the lines  $$ \begin{pmatrix}
xyz \end{pmatrix} =\begin{pmatrix}
t+1
2t+4
-3t +5
\end{pmatrix},\;\;\;\begin{pmatrix}
xyz \end{pmatrix} =\begin{pmatrix}
4t-12
-t+8
t+17
\end{pmatrix}. $$  

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