
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
When constructing triangles with matchsticks, it is well-known that 3 matchsticks can only form one type of triangle, which can be represented by side lengths of 1, 1, 1. 4 matchsticks cannot form a triangle; 5 matchsticks can only form one type of triangle with side lengths of 2, 2, 1; 6 matchsticks can only form one type of triangle with side lengths of 2, 2, 2; 7 matchsticks can form 2 types of triangles with side lengths of 3, 3, 1 and 3, 2, 2; and so on. The question is, how many different triangles can be constructed using 30 matchsticks?

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