
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 \(\left\{x_{n}\right\}\) satisfies:
\[ x_{1}=a, \, x_{2}=b, \, x_{n+2}=3 x_{n+1}+2 x_{n}, \]

if there exists an integer \(k \geq 3\) such that \(x_{k}=2019\), then determine the number of such positive integer ordered pairs \((a, b)\).

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