
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 \( g(1)=0 \), \( 20<g(5)<30 \), \( 40<g(6)<50 \), \( 3000k<g(100)<3000(k+1) \) for some integer \( k \). What is \( k \)?

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