
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 $f(x) = a(x + \frac{1}{x}) - |x - \frac{1}{x}| (x > 0)$, where $a \in \mathbb{R}$.
(1) If $a = \frac{1}{2}$, find the monotonic intervals of $y = f(x)$;
(2) If the equation $f(x) = t$ has four distinct solutions $x\_1$, $x\_2$, $x\_3$, $x\_4$, find the conditions that the real numbers $a$, $t$ should satisfy.

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