
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
The original equation can be represented as:

\[1-2 \operatorname{Sin}(x / 2) \operatorname{Cos}(x / 2)=\frac{\operatorname{Sin}(x / 2)-\operatorname{Cos}(x / 2)}{\operatorname{Cos}(x / 2)} \Leftrightarrow(\operatorname{Sin}(x / 2)-\operatorname{Cos}(x / 2))^{2}=\frac{\operatorname{Sin}(x / 2)-\operatorname{Cos}(x / 2)}{\operatorname{Cos}(x / 2)}\]

Hence, \(\operatorname{Sin}(x / 2)-\operatorname{Cos}(x / 2) = 0\), therefore \(x = \frac{\pi}{2} + 2\pi k\). Dividing by \(\operatorname{Sin}(x / 2)-\operatorname{Cos}(x / 2)\), we obtain the equation: 

\[1 + \operatorname{Cos}^2 \frac{x}{2} = \frac{1}{2} \operatorname{Sin} x\]

This equation has no solutions.

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