
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  $T$  be a right triangle with sides having lengths  $3$ ,  $4$ , and  $5$ . A point  $P$  is called *awesome* if P is the center of a parallelogram whose vertices all lie on the boundary of  $T$ . What is the area of the set of awesome points?

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