
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
Let the quadratic function be $f(x)=x^{2}+ax+a$.  
$(1)$ If the equation $f(x)-x=0$ has two real roots $x_{1}$ and $x_{2}$ that satisfy $0 < x_{1} < x_{2} < 1$, find the range of the real number $a$.  
$(2)$ Find the minimum value of the function $g(x)=af(x)-a^{2}(x+1)-2x$ in the interval $[0,1]$.

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