
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 $y=a\left(x-3\right)^{2}+2$ passes through the point $\left(1,-2\right)$.  
$(1)$ Find the value of $a$.  
$(2)$ If points $A(m$,$y_{1})$ and $B(n$,$y_{2})(m \lt n \lt 3)$ both lie on the parabola, compare the sizes of $y_{1}$ and $y_{2}$.

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