
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
Find the minimum value of $x\_1 + x\_2 + \frac{a}{x\_1 x\_2}$ given that the solution set of the inequality $x^2 - 4ax + 3a^2 < 0$ ($a > 0$) in terms of $x$ is $(x\_1, x\_2)$.

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