
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]

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# Your action step 2

# Your action step 3

...

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Let \( x \) be the number of happy subjects, \( y \) be the number of redheads, and \( z \) be the total number of subjects. According to the given conditions,

\[ 
\frac{40}{100} x = \frac{60}{100} y, \quad z = x + \frac{40}{100} y = x + \frac{2}{5} y 
\]

We have \( 2x = 3y \). Express all variables through a parameter \( t \in \mathbb{N} \) :

\[ 
x = 3t, \quad y = 2t, \quad z = 3t + \frac{4t}{5} = \frac{19}{5} t 
\]

Since \( z \) is a natural number, \( t = 5k, k \in \mathbb{N} \), hence \( x = 15k, y = 10k, z = 19k \). Then, the number of unhappy redheads is \( \frac{2}{5} y = 4k \). The ratio of the number of unhappy redheads to the total number of subjects is

\[ \frac{4k}{19k} = \frac{4}{19} \]

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