
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
If $a$, $b$, and $c$ are the three sides of a right-angled triangle with $c$ being the hypotenuse, then the Pythagorean theorem states that $c^2=a^2+b^2$. We wish to extend this theorem to solid geometry: In the tetrahedron $O-ABC$ with $S$ as the area of face opposite to vertex $O$, and $S_1$, $S_2$, $S_3$ as the areas of the side faces $\triangle AOB$, $\triangle BOC$, $\triangle COA$ respectively, with $OA$, $OB$, $OC$ all mutually perpendicular, then the relationship between $S$ and $S_1$, $S_2$, $S_3$ is __________.

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