
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
Determine the value of \( k \) for which the volume of the parallelepiped formed by the vectors \( \begin{pmatrix} 1 \\ 4 \\ 5 \end{pmatrix}, \begin{pmatrix} 2 \\ k \\ 3 \end{pmatrix}, \) and \( \begin{pmatrix} 1 \\ 3 \\ k \end{pmatrix} \) equals 20, where \( k > 0 \).

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