
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
Let  $a_1, a_2, a_3, a_4$  be integers with distinct absolute values. In the coordinate plane, let  $A_1=(a_1,a_1^2)$ ,  $A_2=(a_2,a_2^2)$ ,  $A_3=(a_3,a_3^2)$  and  $A_4=(a_4,a_4^2)$ . Assume that lines  $A_1A_2$  and  $A_3A_4$  intersect on the  $y$ -axis at an acute angle of  $\theta$ . The maximum possible value for  $\tan \theta$  can be expressed in the form  $\dfrac mn$  for relatively prime positive integers  $m$  and  $n$ . Find  $100m+n$ .

*Proposed by James Lin*

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