
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 $a$ is a constant, and the quadratic function $y=x^{2}-ax+a+3$.  
$(1)$ If the graph of this quadratic function intersects the $x$-axis, find the range of values for $a$;  
$(2)$ Given $y\geqslant 4$, find the solution set for $x$;  
$(3)$ If there exists $x\in \left[2,4\right]$ such that $y=0$, find the range of values for $a$.

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