
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
In a mathematics test number of participants is  $N < 40$ . The passmark is fixed at  $65$ . The test results are
the following: 
The average of all participants is  $66$ , that of the promoted  $71$  and that of the repeaters  $56$ . 
However, due to an error in the wording of a question, all scores are increased by  $5$ . At this point
the average of the promoted participants becomes  $75$  and that of the non-promoted  $59$ .
(a) Find all possible values ​​of  $N$ .
(b) Find all possible values ​​of  $N$  in the case where, after the increase, the average of the promoted had become  $79$  and that of non-promoted  $47$ .

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