
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  $ABCD$  be a convex quadrilateral with positive area such that every side has a positive integer length and  $AC=BC=AD=25$ . If  $P_{max}$  and  $P_{min}$  are the quadrilaterals with maximum and minimum possible perimeter, the ratio of the area of  $P_{max}$  and  $P_{min}$  can be expressed in the form  $\frac{a\sqrt{b}}{c}$  for some positive integers  $a,b,c$ , where  $a,c$  are relatively prime and  $b$  is not divisible by the square of any integer. Find  $a+b+c$ . 

*Proposed by **FedeX333X***

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