
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
If all sides of a triangle are integers and at least one of its internal angles is $\frac{\pi}{3}$, then this triangle is called a "perfect triangle". There are the following propositions about "perfect triangles":
(1) There exists a right-angled triangle that is a "perfect triangle";
(2) There does not exist a "perfect triangle" with an area that is an integer;
(3) The maximum area of a "perfect triangle" with a perimeter of 12 is $4\sqrt{3}$;
(4) If two "perfect triangles" have two corresponding sides equal and their areas are equal, then these two "perfect triangles" are congruent.
The true propositions are \_\_\_\_\_\_. (Write down the numbers of all true propositions.)

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