
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
A function \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) satisfies: \( f(0) = 0 \) and
$$
\left| f \left( (n+1) 2^k \right) - f \left( n 2^k \right) \right| \leq 1
$$
for all integers \( k \geq 0 \) and \( n \). What is the maximum possible value of \( f(2019) \)?

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