
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

A small agricultural cooperative has established a unique 6-digit numbering system to allow consumers to identify the origin of products:

I) The first 2 digits indicate the sub-region, which are 27, numbered from 01 to 27.

II) The next 2 digits refer to the producer, with each sub-region having 40 producers, numbered from 31 to 70 (note that there is a producer 35, for example, in sub-region 01, in 02, in 03, etc., implying 27 producers numbered 35).

III) The last 2 digits relate to the type of product, which are 28, numbered from 71 to 98.

For instance, the number 074197 corresponds to a product from sub-region 07, producer 41 of this sub-region, type 97.

a) Carlos wrote down the number in the wrong sequence, swapping the 3 parts of the code and noting it as 900950. Is it still possible to identify the sub-region, producer, and type of product?

b) In total, how many different codes can be generated to meet the cooperative's conditions?

c) Júnior is a superstitious consumer. He detests the number 15 and does not buy any product that contains it. How many product codes fall into this category?

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