
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
Let the mass of the new alloy be 1, and let the relative masses of the 1st, 2nd, and 3rd alloys in it be denoted by \( x \), \( y \), and \( z \) respectively. The chromium content in the new alloy is \( k \). The system of equations is given by:

\[
\begin{cases}
x + y + z = 1 \\
0.9x + 0.3z = 0.45 \\
0.4x + 0.1y + 0.5z = k
\end{cases}
\]

From the second equation: \( y = \frac{1}{2} - \frac{1}{3}z \). Substituting this into the first equation, we get: \( x = \frac{1}{2} - \frac{2}{3}z \).

Substituting into the third equation: \( k = 0.25 + 0.2z \).

Since \( z \geq 0 \), the minimum value for \( k \) is \( k_{\text{min}} = 0.25 = 25\% \).

For \( x = \frac{1}{2} - \frac{2}{3}z \Leftrightarrow z = \frac{3}{4} - \frac{3}{2}x \),

\( k = 0.25 + 0.2z = 0.4 - 0.3x \).

Since \( x \geq 0 \), the maximum value for \( k \) is \( k_{\text{max}} = 0.4 = 40\% \).

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