
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 a parabola $y=ax^{2}+bx\left(a\neq 0\right)$ passing through point $A\left(3,3\right)$. Points $M(x_{1}$,$y_{1})$ and $N(x_{2}$,$y_{2})$ are two distinct points on the parabola, and satisfy $x_{1} \lt x_{2}$, $x_{1}+x_{2}=2$.<br/>$(1)$ Express $b$ in terms of $a$ algebraically;<br/>$(2)$ When $y_{1}=y_{2}$, find the axis of symmetry of the parabola and the value of $a$;<br/>$(3)$ When $y_{1} \lt y_{2}$, find the range of values for $a$.

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