
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 $\tan \alpha =3$, find the value of the following expressions:

$(1) \dfrac{4\sin \alpha-\cos \alpha}{3\sin \alpha+5\cos \alpha}$

$(2)$ If $\alpha\in[0,\pi]$, find the value of $\sin \alpha+\cos \alpha$

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