
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 function \( y = f(x) \) is defined on the set \( (0, +\infty) \) and takes positive values on it. It is known that for any points \( A \) and \( B \) on the graph of the function, the areas of the triangle \( AOB \) and the trapezoid \( ABH_BH_A \) are equal (where \( H_A \) and \( H_B \) are the bases of the perpendiculars dropped from points \( A \) and \( B \) onto the x-axis; \( O \) is the origin). Find all such functions.

Given \( f(1) = 4 \), find the value of \( f(4) \).

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