
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
A line in the plane of a triangle  $T$  is called an *equalizer* if it divides  $T$  into two regions having equal area and equal perimeter. Find positive integers  $a>b>c,$  with  $a$  as small as possible, such that there exists a triangle with side lengths  $a,b,c$  that has exactly two distinct equalizers.


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