
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 the sides of a triangle $(a > b > c)$ form an arithmetic sequence with difference $d$, then:

$$
\operatorname{tg} \frac{\alpha}{2} \operatorname{tg} \frac{\gamma}{2} = \frac{1}{3}
$$

and

$$
r = \frac{2d}{3\left(\operatorname{tg} \frac{\alpha}{2} - \operatorname{tg} \frac{\gamma}{2}\right)}
$$

where $r$ is the radius of the inscribed circle of the triangle.

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