
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 if there exists a real number $k$ such that the equation
\[
\begin{pmatrix} 1 \\ 4 \end{pmatrix} + t \begin{pmatrix} 5 \\ -9 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} + s \begin{pmatrix} -2 \\ k \end{pmatrix}
\]
does not have any solutions in $t$ and $s$. If so, find $k$.

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