
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 that the vertex of angle $\alpha$ is at the origin, its initial side coincides with the non-negative semi-axis of $x$, and there are two points $A(1,a)$, $B(2,b)$ on its terminal side. If $\cos 2\alpha = \frac{2}{3}$, then find $|a-b|$.

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