
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
In the Cartesian coordinate system, the parabola $y=a(x^{2}-4x+3)\left(a \gt 0\right)$ intersects the $x$-axis at points $A$ and $B$ $(A$ is to the left of $B)$, and intersects the $y$-axis at point $C$. <br/>$(1)$ Find the axis of symmetry of the parabola; <br/>$(2)$ If point $P\left(m,3\right)$ lies on the parabola and to the right of the axis of symmetry, a line $PQ$ perpendicular to the $x$-axis is drawn passing through point $P$ at point $Q$. <br/>① If $PQ=QA$, find the coordinates of point $C$; <br/>② If $PQ \gt QA$, then the range of $m$ is ______.

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