
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
Let  $ a,\ b$  be postive real numbers. For a real number  $ t$ , denote by  $d(t)$  the distance between the origin and the line  $ (ae^t)x \plus{} (be^{ \minus{} t})y \equal{} 1$ .

Let  $ a,\ b$  vary with  $ ab \equal{} 1$ , find the minimum value of  $ \int_0^1 \frac {1}{d(t)^2}\ dt$ .

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