
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  $P(x)=x^2-3x-9$ . A real number  $x$  is chosen at random from the interval  $5\leq x \leq 15$ . The probability that  $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$  is equal to  $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$ , where  $a,b,c,d$  and  $e$  are positive integers and none of  $a,b,$  or  $c$  is divisible by the square of a prime. Find  $a+b+c+d+e$ .

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