
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 angles of a triangle are $\alpha$, $\beta$, and $\gamma$, and

$$
\operatorname{tg} \frac{\alpha}{2}=\frac{5}{6} \quad \text { and } \quad \operatorname{tg} \frac{\beta}{2}=\frac{30}{27},
$$

then if 

$$
a + c = 2b,
$$

what is $\operatorname{tg} \frac{\gamma}{2}$?

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