
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
Consider a rectangle $R$ with sides $x$ and $y$ where $x < y$. Suppose we remove a square with side $x$ from $R$, resulting in a new rectangle $R'$. It is known that if $R'$ is similar to $R$, then $R$ is a "golden rectangle" and $\frac{y}{x} = \frac{1+\sqrt{5}}{2}$. However, suppose $R'$ is not similar to $R$. Perform the same operation on $R'$, removing a square with side equal to the smaller side of $R'$. The result is a rectangle $R''$. Under what conditions is $R''$ similar to $R$?

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