
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 a sequence of positive numbers $\{a\_n\}$ where $a\_1=1$, $a\_2=2$, and $\{\sqrt{a\_n a\_{n+1}}\}$ is a geometric sequence with a common ratio of $\frac{1}{2}$, find the smallest integer $n$ such that the inequality $$\frac{1}{a\_1} + \frac{1}{a\_2} + \dots + \frac{1}{a\_{2n+1}} > 2019$$ holds.

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