
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 the function $f\left( x \right)=2\sin (\omega x+\varphi )\left( \omega \gt 0,\left| \varphi \right|\lt \frac{\pi }{2} \right)$, the graph passes through point $A(0,-1)$, and is monotonically increasing on $\left( \frac{\pi }{18},\frac{\pi }{3} \right)$. The graph of $f\left( x \right)$ is shifted to the left by $\pi$ units and coincides with the original graph. When ${x}_{1}$, ${x}_{2} \in \left( -\frac{17\pi }{12},-\frac{2\pi }{3} \right)$ and ${x}_{1} \ne {x}_{2}$, if $f\left( {x}_{1} \right)=f\left( {x}_{2} \right)$, find $f({x}_{1}+{x}_{2})$.

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