
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
Define a sequence  $\{a_n\}$   by\[a_0=\frac{1}{2},\ a_{n+1}=a_{n}+\frac{a_{n}^2}{2012}, (n=0,\ 1,\ 2,\ \cdots),\]  find  integer  $k$  such that    $a_{k}<1<a_{k+1}.$ (September 29, 2012, Hohhot)

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