
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
On the edge \(A A_{1}\) of cube \(A B C D A_{1} B_{1} C_{1} D_{1}\) with edge length \(3 \sqrt{2}\), point \(K\) is marked. In space, a point \(T\) is marked such that \(T B=7\) and \(T C=\sqrt{67}\). What is the minimum and maximum length of segment \(TK\)? Provide justification for your answer.

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