
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
Determine the least real number  $ a$  greater than  $ 1$  such that for any point  $ P$  in the interior of the square  $ ABCD$ , the area ratio between two of the triangles  $ PAB$ ,  $ PBC$ ,  $ PCD$ ,  $ PDA$  lies in the interval  $ \left[\frac {1}{a},a\right]$ .

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