
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
Vasya has a stick that is 22 cm long. He wants to break it into three parts with integer lengths and form a triangle from the resulting pieces. How many ways can he do this? (Methods that result in identical triangles are considered the same).

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