
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
4. Suppose that  $\overline{A2021B}$  is a six-digit integer divisible by  $9$ . Find the maximum possible value of  $A \cdot B$ .

5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments?

6. Suppose that  $a$  and  $b$  are positive integers such that  $\frac{a}{b-20}$  and  $\frac{b+21}{a}$  are positive integers. Find the maximum possible value of  $a + b$ .


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