
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
Solve the following problems:<br/>$(1)$ Given $sinα=\frac{5}{{13}}$, find the values of $\cos \alpha$ and $\tan \alpha$;<br/>$(2)$ Given $\tan \alpha =2$, find the value of $\frac{1}{{2sinαcosα+{{cos}^2}α}}$.

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