
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
Suppose $a$, $b$, and $c$ are the three sides of a right-angled triangle, where $c$ is the hypotenuse. Then, $c^{2}=a^{2}+b^{2}$, and this theorem is known as the Pythagorean theorem. Now, extend this theorem to solid geometry: In the tetrahedron $O-ABC$, let $S$ be the area of the face opposite to vertex $O$, and $S_{1}$, $S_{2}$, $S_{3}$ be the areas of the side faces $\triangle AOB$, $\triangle BOC$, $\triangle COA$, respectively. If $OA$, $OB$, $OC$ are pairwise perpendicular, then the relationship between $S$, $S_{1}$, $S_{2}$, and $S_{3}$ is \_\_\_\_\_\_.

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