
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
Find all continuously differentiable functions  $f:[0,1]\to(0,\infty)$  such that  $\frac{f(1)}{f(0)}=e$  and  $$ \int^1_0\frac{\text dx}{f(x)^2}+\int^1_0f'(x)^2\text dx\le2. $$  

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