
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 $f(x)=Asin(ωx+\frac{π}{6})\left(A \gt 0,\omega\ \ \gt 0\right)$, which can only satisfy any two of the following three conditions: ① the maximum value of the function $f\left(x\right)$ is $2$; ② the graph of the function $f\left(x\right)$ can be obtained by shifting the graph of $y=\sqrt{2}sin(x-\frac{π}{4})$ along the $x$-axis; ③ the distance between adjacent axes of symmetry of the graph of the function $f\left(x\right)$ is $\frac{π}{2}$.  
$(1)$ Please write down the numbers of these two conditions, find the analytical expression of $f\left(x\right)$, and determine the range of $f\left(x\right)$ on $[0,\frac{π}{4}]$;  
$(2)$ Find the sum of all solutions of the equation $f\left(x\right)+1=0$ on the interval $\left[-\pi ,\pi \right]$.

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