
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
Given a sequence $\{a_n\}$ that satisfies $a_0= \frac{1}{2}$ and $a_{n+1}=a_n+ \frac{a_n^2}{2018}$ for $n=0,1,2,\ldots$, if $a_k < 1 < a_{k+1}$, what is the value of the positive integer $k$?

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