
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 $g(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $g(1)=0$, $10 < g(5) < 20$, $30 < g(6) < 40$, and $3000k < g(100) < 3000(k+1)$ for some integer $k$. Determine the value of $k$.

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