
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
How many different triangles can be formed from: a) 40 matchsticks; b) 43 matchsticks?

We need to find the number of triples of natural numbers \( x, y, z \) such that \( x \leq y \leq z \), \( x + y + z = 40 \), and \( x + y > z \). From these inequalities, it follows that \( z \) can take values satisfying \( 14 \leq z \leq 19 \). If \( z = 19 \), then \( x + y = 21 \), with \( x \leq y \leq 19 \). Therefore, \( 11 \leq y \leq 19 \), giving us 9 triangles with \( z = 19 \). Similarly, it's determined that the number of triangles for which \( z = 18, 17, 16, 15, 14 \) is respectively 8, 6, 5, 3, 2, making a total of 33 triangles. Similarly, the number of triangles with a perimeter of 43 is 44.

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