
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 sequence \(\{a_n\}\) defined by:
\[ a_1 = 1, \quad a_2 = \frac{3}{7}, \]
\[ a_n = \frac{a_{n-2} a_{n-1}}{2 a_{n-2} - a_{n-1}} \quad (n \in \mathbb{Z}_+, n \geq 3), \]
find the value of \(p + q\) when \(a_{2019} = \frac{p}{q}\) for \(p, q \in \mathbb{Z}_+\) and \((p, q) = 1\).

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