
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 a sequence of positive integers $a_1, a_2, a_3, \ldots, a_{100}$, where the number of terms equal to $i$ is $k_i$ ($i=1, 2, 3, \ldots$), let $b_j = k_1 + k_2 + \ldots + k_j$ ($j=1, 2, 3, \ldots$),  
define $g(m) = b_1 + b_2 + \ldots + b_m - 100m$ ($m=1, 2, 3, \ldots$).  
(I) Given $k_1 = 40, k_2 = 30, k_3 = 20, k_4 = 10, k_5 = \ldots = k_{100} = 0$, calculate $g(1), g(2), g(3), g(4)$;  
(II) If the maximum term in $a_1, a_2, a_3, \ldots, a_{100}$ is 50, compare the values of $g(m)$ and $g(m+1)$;  
(III) If $a_1 + a_2 + \ldots + a_{100} = 200$, find the minimum value of the function $g(m)$.

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