
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
Three points  $P_1, P_2,$  and  $P_3$  and three lines  $\ell_1, \ell_2,$  and  $\ell_3$  lie in the plane such that none of the three points lie on any of the three lines. For (not necessarily distinct) integers  $i$  and  $j$  between 1 and 3 inclusive, we call a line  $\ell$   $(i, j)$ -*good* if the reflection of  $P_i$  across  $\ell$  lies on  $\ell_j$ , and call it *excellent* if there are two distinct pairs  $(i_1, j_1)$  and  $(i_2, j_2)$  for which it is good. Suppose that exactly  $N$  excellent lines exist. Compute the largest possible value of  $N$ .

*Proposed by Yannick Yao*

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