
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 function $f(x) = |\sin x|$ for $x \geq 0$ intersects with a line passing through the origin exactly three times, and that the maximum x-coordinate of the intersection points is $\alpha$, find the value of $\frac {(1+\alpha^{2})\sin 2\alpha}{\alpha}$.

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