
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
6.3 "If $r$ and $R$ are the radii of the incircle and circumcircle of a triangle, respectively, then $R \geq 2r$." A stronger form of this inequality is: If $r$ and $R$ are the radii of the incircle and circumcircle of a triangle, respectively, and $d$ is the distance between the incenter and the circumcenter, then $d^{2} = R^{2} - 2Rr$. Please explain the relationship between these two statements and provide a proof.

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