
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
Given that the area of triangle $\triangle ABC$ is $S$, points $D$, $E$, and $F$ are the midpoints of $BC$, $CA$, and $AB$ respectively. $I_{1}$, $I_{2}$, and $I_{3}$ are the incenters of triangles $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ respectively. Determine the value of the ratio $S^{\prime} / S$, where $S^{\prime}$ is the area of triangle $\triangle I_{1}I_{2}I_{3}$.

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