
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 that the quadratic function $y=ax^{2}+bx-1$ intersects the linear function $y=ax$ at the point $\left(-2,1\right)$.  
$(1)$ Find the values of $a$ and $b$;  
$(2)$ If the points $A(2$,$y_{1})$, $B(b$,$y_{2})$, $C(a-b$,$y_{3})$ lie on the parabola, compare the values of $y_{1}$, $y_{2}$, and $y_{3}$, and explain the reason.

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