
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 $0 < \alpha < \frac{\pi}{2}, 0 < \beta < \frac{\pi}{2}, \cos \alpha= \frac{3}{5}, \cos (\beta+\alpha)= \frac{5}{13}$.  
$(I)$ Find the value of $\sin \beta$;  
$(II)$ Find the value of $\frac{\sin 2\alpha}{\cos^2\alpha +\cos 2\alpha}$.

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