
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 vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\) such that \(|\boldsymbol{a} - \boldsymbol{b}| = 3\), \(|\boldsymbol{a} + 2\boldsymbol{b}| = 6\), and \(\boldsymbol{a}^2 + \boldsymbol{a} \cdot \boldsymbol{b} - 2\boldsymbol{b}^2 = -9\), find \(|\boldsymbol{b}|\).

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