
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 the function $f(x)=x^{2}-2ax+a+2$, where $a\in\mathbb{R}$.
(1) If the equation $f(x)=0$ has two unequal real roots both less than $2$, find the range of values for the real number $a$;
(2) If the inequality $f(x)\geqslant -1-ax$ holds for any $x\in\mathbb{R}$, find the range of values for the real number $a$;
(3) If the maximum value of the function $f(x)$ on $[0,2]$ is $4$, find the value of the real number $a$.

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