
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 $c_0 = 3, d_0 = -1$, and for $n \geq 0$, let
\[
c_{n+1} = c_n + d_n + 2\sqrt{c_n^2 + d_n^2}, \quad d_{n+1} = c_n + d_n - 2\sqrt{c_n^2 + d_n^2}.
\]
Find $\frac{1}{c_{2013}} + \frac{1}{d_{2013}}.$

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