
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 $A A^{\prime}$ of a cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ with edge length 2, a point $K$ is marked. A point $T$ is marked in space such that $T B = \sqrt{11}$ and $T C = \sqrt{15}$. Find the length of the height of the tetrahedron $T B C K$ dropped from the vertex $C$.

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