
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
Let \(\mathbb{N}\) be the set of positive integers, and let \(f: \mathbb{N} \rightarrow \mathbb{N}\) be a function satisfying:
- \(f(1) = 1\)
- For \(n \in \mathbb{N}\), \(f(2n) = 2f(n)\) and \(f(2n + 1) = 2f(n) - 1\).

Determine the sum of all positive integer solutions to \(f(x) = 19\) that do not exceed 2019.

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