
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  $\lfloor x\rfloor$  denote the greatest integer function and  $\{x\}=x-\lfloor x\rfloor$  denote the fractional part of  $x$ .  Let  $1\leq x_1<\ldots<x_{100}$  be the  $100$  smallest values of  $x\geq 1$  such that  $\sqrt{\lfloor x\rfloor\lfloor  x^3\rfloor}+\sqrt{\{x\}\{x^3\}}=x^2.$  Compute \[\sum_{k=1}^{50}\dfrac{1}{x_{2k}^2-x_{2k-1}^2}.\]

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