
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 expression $\eta(x)-\eta(x-7\pi) = \left\{\begin{array}{l}1, \quad x \in [0,7\pi) \\ 0, \quad x \in (-\infty, 0) \cup [7\pi, +\infty) \end{array}\right.$

Case 1. \(x \in [0, 7\pi)\). The equation takes the form

\[ \sin x = 1 + \cos x \]

Applying the method of introduction of an auxiliary argument, we get

\[ \sin \left(x - \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

Solving this equation, we get

\[ \left\{\begin{array}{l} x = \frac{\pi}{2} + 2\pi k, \quad k \in \mathbb{Z} \\ x = \pi + 2\pi m, \quad m \in \mathbb{Z} \end{array}\right. \]

Considering \(x \in [0, 7\pi)\), we have

\[ \left\{\begin{array}{l} x = \frac{\pi}{2} + 2\pi k, \quad k = 0, 1, 2, 3 \\ x = \pi + 2\pi m, \quad m = 0, 1, 2 \end{array}\right. \]

Case 2. \(x \in (-\infty, 0) \cup [7\pi, +\infty)\). The equation takes the form

\[ 1 + \cos x = 0 \]

Solving this equation, we get

\[ x = \pi + 2\pi m, \quad m \in \mathbb{Z} \]

Considering \(x \in (-\infty, 0) \cup [7\pi, +\infty)\), we have

\[ x = \pi + 2\pi m, \quad m = -1, -2, -3, \ldots, 3, 4, \ldots \]

Combining the solutions obtained in the above cases, we find \( x = \frac{\pi}{2} + 2\pi k, \quad k = 0, 1, 2, 3;\quad x = \pi(2m + 1), \quad m \in \mathbb{Z} \).

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