
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 \( c_0 = -3, d_0 = 2 \), and for \( n \geq 0 \), let
\begin{align*}
c_{n+1} &= 2c_n + d_n + \sqrt{c_n^2 + d_n^2}, \\
d_{n+1} &= 2c_n + d_n - \sqrt{c_n^2 + d_n^2}.
\end{align*}
Find \( \frac{1}{c_{10}} + \frac{1}{d_{10}} \).

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