
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
A rhombus  $\mathcal R$  has short diagonal of length  $1$  and long diagonal of length  $2023$ . Let  $\mathcal R'$  be the rotation of  $\mathcal R$  by  $90^\circ$  about its center. If  $\mathcal U$  is the set of all points contained in either  $\mathcal R$  or  $\mathcal R'$  (or both; this is known as the *union* of  $\mathcal R$  and  $\mathcal R'$ ) and  $\mathcal I$  is the set of all points contained in both  $\mathcal R$  and  $\mathcal R'$  (this is known as the *intersection* of  $\mathcal R$  and  $\mathcal R'$ , compute the ratio of the area of  $\mathcal I$  to the area of  $\mathcal U$ .

*Proposed by Connor Gordon*

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