
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  $ABC$  be a triangle with sides  $51, 52, 53$ . Let  $\Omega$  denote the incircle of  $\bigtriangleup ABC$ . Draw tangents to  $\Omega$  which are parallel to the sides of  $ABC$ . Let  $r_1, r_2, r_3$  be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed  $r_1 + r_2 + r_3$ .

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