
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 vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\) satisfy the following conditions:
$$
\begin{array}{l}
|a - b| = 3, \\
|a + 2b| = 6, \\
a^{2} + a \cdot b - 2b^{2} = -9.
\end{array}
$$

Find the magnitude of \(\boldsymbol{b}\), \(|b| =\).

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