
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 \( h \) is the height of a triangle, \( b \) is the base, and \( x \) is the smaller of its segments, then the radius of the circumcircle according to Vydman's formula is expressed as:

$$
r = \sqrt{\binom{b}{2}^{2} + \left( \frac{h^{2} + \left( \frac{b}{2} - x \right)^{2} - \frac{b^{2}}{4}}{2h} \right)^{2}}
$$

Verify the validity of this formula.

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