
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 $α∈({0,\frac{π}{4}})$, $tan({α+\frac{π}{4}})=2cos2α$.  
$(1)$ Find the value of $α$;  
$(2)$ Let $f\left(x\right)=\sin \left(x+2\alpha \right)$, $x\in \left[0,\pi \right]$. Find the monotonic intervals and range of $f\left(x\right)$.

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