
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 the matrix $A= \begin{pmatrix} 1 & 2 & -1 \\ 2 & 2 & -3\\end{pmatrix}$, and matrix $B= \begin{pmatrix} a \\ -2a \\ 3a\\end{pmatrix}$. If $AB= \left(\begin{matrix} 12 \\ 22\\end{matrix}\right.)$, then $a=$ \_\_\_\_\_\_ .

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