
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
The solution to the equation $f\left( \left. x \right. \right)=x$ is called the fixed point of the function $f\left( \left. x \right. \right)$. If $f\left( \left. x \right. \right)= \frac{ax}{x+1}$ has a unique fixed point, and the sequence $\{a_{n}\}$ satisfies $a_{1}=1$, $\frac{1}{a_{n+1}}=f\left( \left. \frac{1}{a_{n}} \right. \right)$, then $a_{2019}=$\_\_\_\_\_\_\_\_\_\_.

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