
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
In a square lawn with a side length of $a$, there are sprinklers installed at each of the four corners. Each sprinkler can rotate and spray water over a $90^{\circ}$ angle, reaching from one side of the corner to the other, with an effective range of $a$. What proportion of the lawn can be simultaneously covered by the water from all four sprinklers?

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