
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 that $\tan \alpha=2$, find the value of the following expressions:
$$(1) \frac{\sin \alpha+2\cos \alpha}{4\cos \alpha-\sin \alpha}$$
$$(2) \sqrt{2}\sin(2\alpha+\frac{\pi}{4})+1$$

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