
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 sequence $\{a_n\}$ satisfies $a_1= \frac{1}{2}$, and $a_n \cdot a_{n-1} - 2a_n + a_{n-1} = 0$ $(n \geqslant 2)$, then the largest positive integer $k$ that makes $a_k > \frac{1}{2,017}$ is ______.

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