
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 and  $I$  its incenter. Suppose  $AI=\sqrt{2}$ ,  $BI=\sqrt{5}$ ,  $CI=\sqrt{10}$  and the inradius is  $1$ . Let  $A'$  be the reflection of  $I$  across  $BC$ ,  $B'$  the reflection across  $AC$ , and  $C'$  the reflection across  $AB$ . Compute the area of triangle  $A'B'C'$ .

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