
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
Assume that \( T \) is the period of the function \( y = f(x) \). From the given condition we have:

$$
\sin x = f(x) - 0.4 f(x - \pi) = f(x - T) - 0.4 f(x - T - \pi) = \sin (x - T)
$$

Since this equality holds for all \( x \), the number \( T \) is a period of the function \( y = \sin x \), that is, \( T = 2 \pi n \), where \( n \) is a natural number. From this equation, we get:

$$
\begin{gathered}
f(x) = 0.4 f(x - \pi) + \sin x = 0.4 (0.4 f(x - 2 \pi) - \sin x) + \sin x = \\
= 0.4 (0.4 (0.4 f(x - 3 \pi) + \sin x) - \sin x) + \sin x = \\
= \ldots = 0.4 (0.4 (\ldots(0.4 f(x - 2 \pi n) - \sin x) + \ldots + \sin x) - \sin x) + \sin x = \\
= 0.4^{2n} f(x) - \sin x \left( 0.4^{2n - 1} - 0.4^{2n - 2} + 0.4^{2n - 3} - \ldots + 0.4 - 1 \right).
\end{gathered}
$$

Then

$$
f(x) \left( 1 - 0.4^{2n} \right) = \sin x \left( 1 - 0.4 + 0.4^2 - \ldots - 0.4^{2n - 1} \right) = \sin x \frac{1 - 0.4^{2n}}{1 + 0.4}.
$$

Therefore, \( f(x) = \frac{5}{7} \sin x \).

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