
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 \( f(2) = 0 \), \( 100 < f(7) < 110 \), \( 120 < f(8) < 130 \), and \( 6000k < f(100) < 6000(k+1) \) for some integer \( k \). Determine the value of \( k \).

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