
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
Given that \begin{align*}x &= 1 - \frac 12 + \frac13 - \frac 14 + \cdots + \frac1{2007}, y &= \frac{1}{1005} + \frac{1}{1006} + \frac{1}{1007} + \cdots + \frac 1{2007},\end{align*}
find the value of  $k$  such that \[x = y + \frac 1k.\]

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