
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 $f(x) = x + \frac{a}{2x}$, two tangent lines are drawn through the point $(1, 0)$, with tangent points $A(x\_1, f(x\_1))$ and $B(x\_2, f(x\_2))$, where $0 < x\_1 < x\_2$. Find the range of $a$ such that there exists a unique integer in the interval $(x\_1, x\_2)$.

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