
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
The function $f(x)=|\sin x|$ and the line $y=kx$ $(k > 0)$ have exactly three intersection points, and the maximum value of the x-coordinates of the intersection points is $\alpha$. Then, $\dfrac{\cos \alpha}{\sin \alpha+\sin 3\alpha}=$ __________ (expressed in terms of $\alpha$).

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