
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
A sequence \( (c_1,d_1) \), \( (c_2,d_2) \), \( (c_3,d_3) \), \(\ldots\) of points in the coordinate plane satisfies
\[
(c_{n + 1}, d_{n + 1}) = (\sqrt {2}c_n - d_n, \sqrt {2}d_n + c_n)
\]
for \( n = 1,2,3,\ldots \). Suppose that \( (c_{50},d_{50}) = (1,3) \). What is \( c_1 + d_1 \)? Express your answer using exponential notation.

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