
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
Given a non-zero 3rd order matrix $B$ ($B\neq O$, where $O$ is the zero matrix), and each column vector of $B$ is a solution to the system of equations $\begin{cases} x+2y-2z=0 \\ 2x-y+\lambda z=0 \\ 3x+y-z=0 \end{cases}$, find the value of $\lambda$.

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