
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 plane rectangular coordinate system $xOy$, points $A(-3, y_{1})$, $B(\frac{a}{2}, y_{2})$, $C(m, y_{3})$ lie on the parabola $y=-x^{2}+2ax+c\left(a \gt 0\right)$. <br/>$(1)$ The axis of symmetry of the parabola is the line $x=$______, write down the relationship between $y_{1}$ and $y_{2}$ as $y_{1}$______$y_{2}$; <br/>$(2)$ If $m=4$ and $y_{1}=y_{3}$, then the value of $a$ is ______; <br/>$(3)$ If for any $1\leqslant m\leqslant 4$, we have $y_{1} \lt y_{3} \lt y_{2}$, find the range of values for $a$.

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