
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 $f(x) = \begin{cases} x + \frac{1}{x}, & x \in \left[-1, -\frac{1}{2}\right), \\ -\frac{5}{2}, & x \in \left[-\frac{1}{2}, \frac{1}{2}\right), \\ x - \frac{1}{x}, & x \in \left[\frac{1}{2}, 1\right]. \end{cases}$

(1) Find the range of $f(x)$.

(2) Let $g(x) = ax - 3$, where $x \in [-1, 1]$. If for any $x\_1 \in [-1, 1]$, there exists $x\_0 \in [-1, 1]$ such that $g(x\_0) = f(x\_1)$, find the range of real values for $a$.

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