
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 $(1+i)^{2010} = 1 + \binom{2010}{1}i - \binom{2010}{2} - \binom{2010}{3}i + \ldots + \binom{2010}{k}i^k + \ldots - \binom{2010}{2010}$ (where $i$ is the imaginary unit), it can be deduced that $\binom{2010}{1} - \binom{2010}{3} + \binom{2010}{5} - \ldots + (-1)^k\binom{2010}{2k+1} + \ldots + \binom{2010}{2009} = \boxed{}$.

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