
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 a line \(l\) passing through the origin and the function \(y = |\sin x|\) for \(x \geq 0\), such that they intersect at exactly three points, let \(\alpha\) be the largest x-coordinate of these intersection points. Find the value of 
\[
\frac{\left(1+\alpha^{2}\right) \sin 2 \alpha}{2 \alpha}.
\]

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