
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
A triangle $ABC$ has a base $AB$ equal to $\frac{\sqrt{3}}{2}$ and an altitude $CH$ dropped to this base which is equal to $\frac{\sqrt{6}}{3}$. It is known that point $H$ lies on $AB$ and $AH : HB = 2 : 1$. A circle is inscribed in the angle $ABC$ of triangle $ABC$, and its center lies on altitude $CH$. Find the radius of this circle.

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