
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 the function $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are integers, and the conditions $f(1) = 0, 50 < f(7) < 60, 70 < f(8) < 80,$ and $5000k < f(100) < 5000(k+1)$ for some integer $k$, determine the value of $k$.

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