
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
The matrix
\[
\begin{pmatrix} 
\frac{1}{18} & -\frac{2}{9} \\ 
-\frac{2}{9} & \frac{8}{9} 
\end{pmatrix}
\]
corresponds to projecting onto a certain vector \( \begin{pmatrix} a \\ b \end{pmatrix} \). Find \( \frac{b}{a} \).

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