
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 all real numbers  $a>0$  for which there exists a nonnegative continuous function  $f(x)$  defined on  $[0,a]$  with the property that the region
 $R=\{(x,y): 0\le x\le a, 0\le y\le f(x)\}$ 
has perimeter  $k$  units and area  $k$  square units for some real number  $k$ .

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