
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
We know that in rectangle $ABCD$, if we set $AB=a$ and $BC=b$, then the radius $R$ of the circumscribed circle of rectangle $ABCD$ satisfies: $4R^{2}=a^{2}+b^{2}$. By analogy with the above conclusion, answer the following: In the rectangular solid $ABCD-A_{1}B_{1}C_{1}D_{1}$, if we set $AB=a$, $AD=b$, and $AA_{1}=c$, then the relationship that the radius $R$ of the circumscribed sphere of the rectangular solid $ABCD-A_{1}B_{1}C_{1}D_{1}$ satisfies is ______________.

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