
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
Suppose that \( f_{1}(x)=\frac{1}{2-x} \). For each positive integer \( n \geq 2 \), define \( f_{n}(x)=f_{1}\left(f_{n-1}(x)\right) \) for all real numbers \( x \) in the domain of \( f_{1}\left(f_{n-1}(x)\right) \). The value of \( f_{2019}(4) \) can be written as \( \frac{a}{b} \) where \( a \) and \( b \) are positive integers with no common divisor larger than 1. What is \( (a, b) \)?

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