
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 axis of symmetry of the parabola $y=x^{2}+bx+c$ is the line $x=2$.  
$(1)$ Find the value of $b$;  
$(2)$ When $1\leqslant x\leqslant 4$, the sum of the maximum and minimum values of the function $y$ is $6$. Find the value of $c$;  
$(3)$ When $1 \lt x \lt 4$, the parabola intersects the $x$-axis at exactly one point. Find the range of values for $c$.

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