
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 $y$ is a quadratic function of $x$ in the form of $y=ax^{2}-bx+2 (a\ne 0).$

(1) When $a=-2$, $b=-4$, find the axis of symmetry and the vertex coordinates of the function graph.

(2) Under the conditions in (1), $Q(m,t)$ is a point on the graph, and its symmetric point $P$ about the origin also falls on the graph. Find the value of $m$.

(3) When the graph of the function passes through point $(1,0)$, if $A(\frac{1}{2},y_{1})$ and $B(\frac{1}{2}+\frac{3}{a},y_{2})$ are two points on the graph, compare the sizes of $y_{1}$ and $y_{2}$.

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