
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 function $y=f(x)$ is an even function defined on the domain $\mathbb{R}$, and for $x \geqslant 0$, 
$$
f(x)= 
\begin{cases} 
\left( \frac {1}{2}\right)^{x}+1, & (x > 2) \\
\frac {5}{16}x^{2}, & (0\leqslant x\leqslant 2)
\end{cases}
$$
Determine the range of the real number $a$ for the equation $[f(x)]^{2}+af(x)+b=0$ which has exactly $6$ distinct real roots.

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