
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
In a right triangle $ABC$, where $\angle C=90^{\circ}$, $AC=b$, $BC=a$, the radius $r$ of the circumcircle of $\triangle ABC$ is $\frac{\sqrt{a^2+b^2}}{2}$. By analogy, if the three lateral edges of a triangular pyramid are mutually perpendicular and their lengths are $a, b, c$ respectively, then the radius $R$ of its circumscribed sphere is \_\_\_\_\_\_.

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