
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
Given two linear functions \( f(x) \) and \( g(x) \) such that the graphs \( y = f(x) \) and \( y = g(x) \) are parallel lines, not parallel to the coordinate axes. It is known that the graph of the function \( y = (f(x))^2 \) touches the graph of the function \( y = 4 g(x) \). Find all values of \( A \) such that the graph of the function \( y = (g(x))^2 \) touches the graph of the function \( y = A f(x) \).

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