
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 graph of the quadratic function $y=ax^{2}+bx+c$ passes through the points $A(-2,9)$, $B(0,3)$, and $C(4,3)$.

(1) Find the relationship of this quadratic function and determine the coordinates of its vertex $M$.

(2) If points $P(m,y_{1})$ and $Q(m+1,y_{2})$ both lie on the graph of this function, compare the values of $y_{1}$ and $y_{2}$.

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