
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 parabola $y=-x^{2}+bx-3$ passes through the point $\left(-1,0\right)$.  
$(1)$ Find the value of $b$;  
$(2)$ Find the coordinates of the vertex;  
$(3)$ If there are two points $(1$,$y_{1})$ and $(-1$,$y_{2})$ on the parabola, then $y_{1}$______$y_{2}$.

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