
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
Example: (1) Use ● to represent a solid circle and ○ to represent a hollow circle. Now, several solid and hollow circles are arranged in a certain pattern as follows:  
●○●●○●●●○●○●●○●●●○●○●●○●●●○…  
Question: In the first 2001 circles, there are     hollow circles.  
(2) Ancient Greek mathematicians called the numbers 1, 3, 6, 10, 15, 21, … triangle numbers, which have a certain pattern. What is the difference between the 24th triangle number and the 22nd triangle number?

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