
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
**p19.**  $A_1A_2...A_{12}$  is a regular dodecagon with side length  $1$  and center at point  $O$ . What is the area of the region covered by circles  $(A_1A_2O)$ ,  $(A_3A_4O)$ ,  $(A_5A_6O)$ ,  $(A_7A_8O)$ ,  $(A_9A_{10}O)$ , and  $(A_{11}A_{12}O)$ ? $(ABC)$  denotes the circle passing through points  $A,B$ , and  $C$ .**p20.** Let  $N = 2000... 0x0 ... 00023$  be a  $2023$ -digit number where the  $x$  is the  $23$ rd digit from the right. If $ N$  is divisible by  $13$ , compute  $x$ .**p21.** Alice and Bob each visit the dining hall to get a grilled cheese at a uniformly random time between  $12$  PM and  $1$  PM (their arrival times are independent) and, after arrival, will wait there for a uniformly random amount of time between  $0$  and  $30$  minutes. What is the probability that they will meet?


PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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