
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

At Galbraith H.S., the lockers are arranged in banks of 20 lockers. Each bank of lockers consists of six columns of lockers; the first two columns in each bank consist of two larger lockers and the last four columns in each bank consist of four smaller lockers. The lockers are numbered consecutively starting at 1, moving down each column and then down the next column, and so on.  

(a) What is the sum of the locker numbers of the column of lockers that contain the number 24?

(b) The sum of the locker numbers for one column is 123. What are the locker numbers in this column?

(c) The sum of the locker numbers for another column is 538. What are the locker numbers in this column?

(d) Explain why 2013 cannot be a sum of any column of locker numbers.

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