
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 $0 < \alpha < \frac{π}{2}$, $0 < \beta < \frac{π}{2}$, $\sin \alpha = \frac{4}{5}$, $\cos (\alpha + \beta) = \frac{5}{13}$. Find:

$(1)$ The value of $\cos \beta$;

$(2)$ The value of $\frac{{\sin^2 \alpha + \sin 2\alpha}}{{\cos 2\alpha - 1}}$.

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