[Example 4.1.1] There is a square piece of paper, and inside the paper, there are $n$ different points. Now, let $M$ represent the set of points consisting of the four vertices of the square and these $n$ points, totaling $n+4$ points. The task is to cut this square piece of paper into some triangles according to the following rules:
(1) Each triangle's 3 vertices are elements of the point set $M$;
(2) Apart from the vertices of the triangles, no triangle contains any points from the set $M$.
How many triangles can be cut out in total? How many cuts are needed in total? (Each cut to form one side of a triangle counts as one cut.)
When solving this problem, it is evident that considering each triangle individually is not feasible. Therefore, we must think about the triangles as a whole. Since this problem does not require discussing "how to cut," but only needs to determine "how many triangles are cut out in total," we can focus on the relationship between the number of points \( n \) and the number of triangles. Given that the vertices of the square and each point inside the square can all be vertices of the triangles, we must study the overall characteristics of these points and triangles. For any point inside the square, regardless of how many triangles it forms with other points, the sum of the interior angles formed by the edges extending from this point is always \( 360^\circ \). The sum of the interior angles formed by the edges extending from the vertices of the square is always \( 90^\circ \). This insight leads us to consider the sum of the interior angles of the triangles and the overall sum of the interior angles of all triangles.

Since the sum of the interior angles of each triangle is \( 180^\circ \), the sum of the interior angles formed by the edges extending from each point is \( 360^\circ \), and the sum of the interior angles formed by the edges extending from the vertices of the square is \( 90^\circ \), we can derive the following equation:
\[
n \times 360^\circ + 4 \times 90^\circ = 2 n \times 180^\circ + 2 \times 180^\circ = (2 n + 2) \times 180^\circ.
\]

Therefore, \( 2 n + 2 \) triangles can be cut out.
With this experience, for the problem of "how many cuts," we can think about the number of sides of the triangles as a whole.

Since each triangle has 3 sides, the total number of sides of these triangles is \((2 n + 2) \times 3 = 6 n + 6\). The 4 sides of the square, although they are sides of these triangles, do not need to be cut, so they should be excluded when calculating the number of cuts. Additionally, each side of the triangles inside the square is a common side of two triangles, so one cut produces two sides of two triangles. Therefore, the total number of cuts is
\[
\frac{6 n + 6 - 4}{2} = 3 n + 1 \text{ (cuts). }
\]

Thus, for \( n \) points inside the square, \( 2 n + 2 \) triangles can be cut out, and a total of \( 3 n + 1 \) cuts are made.

From the process of solving this problem, we can see that focusing on the local aspects, such as studying each triangle individually, is difficult and unnecessary. Instead, we can start from the whole, considering the seemingly independent but actually closely related quantities: the total sum of the interior angles of all triangles and the total sum of the sides of all triangles. By gaining an overall understanding of these quantities, we can solve the problem.

When solving math problems, we often encounter situations where it is impossible or unnecessary to focus on the individual components of the problem. Instead, we place the problem in a broader context, grasp it as a whole, observe it as a whole, and handle it as a whole. The overall strategy for solving problems generally includes the following aspects:
1. By summing or multiplying the whole, we obtain local conclusions.
2. By grasping the overall conclusion, we seek a solution path.
3. By supplementing the local parts to form a whole, we derive local results from overall thinking.
4. By considering the overall nature of the conditions, we control the solution process as a whole.
5. By seizing the invariance of the whole, we solve problems from the overall characteristics.
The overall strategy for solving problems generally follows the following思路:

(Note: The last sentence "The overall strategy for solving problems generally follows the following思路:" is incomplete in the original text and has been translated as is. The term "思路" means "line of thought" or "approach" in English.)