
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
On two parallel lines, the distinct points  $A_1,A_2,A_3,\ldots $  respectively  $B_1,B_2,B_3,\ldots $  are marked in such a way that  $|A_iA_{i+1}|=1$  and  $|B_iB_{i+1}|=2$  for  $i=1,2,\ldots $ . Provided that  $A_1A_2B_1=\alpha$ , find the infinite sum  $\angle A_1B_1A_2+\angle A_2B_2A_3+\angle A_3B_3A_4+\ldots $ 

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