
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 the function \( f(x) \) be defined on \(\mathbf{R}\). For any \( x \in \mathbf{R} \), the equations \( f(1+4x)=f(3-2x) \) and \( f(2+3x)=-f(7-6x) \) hold. Find the value of \(\sum_{k=1}^{100}[f(2k-1)+f(4k-2)]\).

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