
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  $N$  be the number of ways of distributing  $52$  identical balls into  $4$  distinguishable boxes such that no box is empty and the difference between the number of balls in any two of the boxes is not a multiple of  $6$  If  $N=100a+b$ , where  $a,b$  are positive integers less than  $100$ , find  $a+b.$ 

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