
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 real number $k'$ such that the following equation has no solutions in $t$ and $s$:
\[\begin{pmatrix} 5 \\ 7 \end{pmatrix} + 2t \begin{pmatrix} 3 \\ -5 \end{pmatrix} = \begin{pmatrix} 4 \\ -1 \end{pmatrix} + s \begin{pmatrix} -2 \\ k' \end{pmatrix}\]

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