
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 $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1) = 0$, $60 < f(9) < 70$, $90 < f(10) < 100$, $10000k < f(100) < 10000(k+1)$ for some integer $k$. What is $k$?

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