
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  $\angle BAC = 90^o$  and  $D$  be the point on the side  $BC$  such that  $AD \perp BC$ . Let $ r, r_1$ , and  $r_2$  be the inradii of triangles  $ABC, ABD$ , and  $ACD$ , respectively. If  $r, r_1$ , and  $r_2$  are positive integers and one of them is  $5$ , find the largest possible value of    $r+r_1+ r_2$ .

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