
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
Calculate the value of the following expressions, given that \(\alpha + \beta + \gamma = 180^\circ\).

\[ \sin \frac{\alpha + \beta}{2} - \cos \frac{\gamma}{2} \]

\[ 
\begin{aligned}
& \operatorname{tg} \frac{\gamma}{2} + \operatorname{tg} \frac{\alpha + \beta}{2} - \left(\operatorname{ctg} \frac{\alpha + \beta}{2} + \operatorname{ctg} \frac{\gamma}{2}\right) \\
& \sin^2 \frac{\alpha + \beta}{2} + \operatorname{ctg} \frac{\alpha + \beta}{2} \operatorname{ctg} \frac{\gamma}{2} - \cos^2 \frac{\gamma}{2}
\end{aligned}
\]

\[ \cos^2 \frac{\alpha + \beta}{2} + \operatorname{tg} \frac{\alpha + \beta}{2} \operatorname{tg} \frac{\gamma}{2} + \cos^2 \frac{\gamma}{2} \]

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