
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 $f(x)= \begin{cases} \frac{1}{2}x^2 + 2x + 2, & x \leqslant 0, \\ | \log_2 x |, & x > 0, \end{cases}$, if the equation $f(x) = a$ has four distinct solutions $x_1$, $x_2$, $x_3$, and $x_4$, where $x_1 < x_2 < x_3 < x_4$, determine the range of values for $\frac{x_1 + x_2}{x_4} + \frac{1}{x_3^2 x_4}$.

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