
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 \( x = \sqrt{1 + \frac{1}{1^{2}} + \frac{1}{2^{2}}} + \sqrt{1 + \frac{1}{2^{2}} + \frac{1}{3^{2}}} + \cdots + \sqrt{1 + \frac{1}{2012^{2}} + \frac{1}{2013^{2}}} \). Find the value of \( x - [x] \), where \( [x] \) denotes the greatest integer not exceeding \( x \).

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