
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
We are given a circle  $ k$  and nonparallel tangents  $ t_1,t_2$  at points  $ P_1,P_2$  on  $ k$ , respectively. Lines  $ t_1$  and  $ t_2$  meet at  $ A_0$ . For a point  $ A_3$  on the smaller arc  $ P_1 P_2,$  the tangent  $ t_3$  to  $ k$  at  $ P_3$  meets  $ t_1$  at  $ A_1$  and  $ t_2$  at  $ A_2$ . How must  $ P_3$  be chosen so that the triangle  $ A_0 A_1 A_2$  has maximum area?

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