
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 non-zero vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ form an angle of $120^{\circ}$. If the vectors $\boldsymbol{a} - \boldsymbol{b}$ and $\boldsymbol{a} + 2 \boldsymbol{b}$ are perpendicular, then find the value of $\left|\frac{\boldsymbol{2a} - \boldsymbol{b}}{\boldsymbol{2a} + \boldsymbol{b}}\right|= $.

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