
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 certain shopping mall designed a lottery promotion as follows: based on different amounts of consumption, during each lottery draw, a customer can draw any number of tickets from 100 tickets with face values of $1, $2, $3, \ldots, $100 (integer face values), and the sum of the face values of the drawn tickets will be the discount amount. A customer was given the opportunity to draw 5 tickets, and Xiao Ming wants to know how many different discount amounts the customer can have.

Modeling the Problem:
Given $n$ integers $1, 2, 3, \ldots, n$ (where $n$ is an integer and $n \geq 3$), how many different results are there when choosing $a$ integers ($1 < a < n$) and summing them up?

Model Exploration:
We adopt a strategy of generalizing from specific problems, starting with the simplest cases and then progressing step by step to find a method to solve the problem.

Exploration 1:
(1) Choosing 2 integers from 1, 2, 3 results in 3, 4, 5 as the possible sums, which are consecutive integers from 3 to 5, resulting in 3 different outcomes.
(2) Choosing 2 integers from 1, 2, 3, 4 results in 3, 4, 5, 6, 7 as the possible sums, which are consecutive integers from 3 to 7, resulting in 5 different outcomes.
(3) Choosing 2 integers from 1, 2, 3, 4, 5 results in ____ different outcomes.
(4) Choosing 2 integers from 1, 2, 3, ..., n (where n is an integer and n ≥ 3) results in ____ different outcomes.

Exploration 2:
(1) Choosing 3 integers from 1, 2, 3, 4 results in ____ different outcomes.
(2) Choosing 3 integers from 1, 2, 3, ..., n (where n is an integer and n ≥ 4) results in ____ different outcomes.

Exploration 3:
Choosing 4 integers from 1, 2, 3, ..., n (where n is an integer and n ≥ 5) results in ____ different outcomes.

General Conclusion:
Choosing $a$ integers ($1 < a < n$) from 1, 2, 3, ..., n (where n is an integer and n ≥ 3) results in ____ different outcomes.

Problem Solution:
Drawing 5 tickets from 100 tickets with face values of $1, $2, $3, \ldots, $100 results in ____ different discount amounts.

Extension:
(1) Choosing 6 integers from 1, 2, 3, ..., n (where n is an integer and n ≥ 6) to have 2023 different sum results. (Write out the solution process)
(2) Choosing $a$ integers ($1 < a < n+1$) from 3, 4, 5, ..., n+3 (where n is an integer and n ≥ 2) results in ____ different sum outcomes.

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