
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

From the given sequence \(a_{1}, a_{2}, a_{3}, \ldots\), a new sequence \(b_{1}, b_{2}, b_{3}, \ldots\) is created as follows, continuing this pattern further based on recognized regularities:

\[ 
\begin{array}{llll}
b_{1}=a_{1}, & b_{2}=a_{2}-a_{1}, & b_{3}=a_{2}, & b_{4}=a_{2}+a_{1}, \\
b_{2}=a_{2}-a_{2}, & b_{6}=a_{3}-a_{1}, & b_{7}=a_{3}, & b_{8}=a_{3}+a_{1} \\
b_{9}=a_{3}+a_{2}, & b_{10}=a_{4}-a_{3}, & b_{11}=a_{4}-a_{2}, & \ldots
\end{array}
\]

Find a method to express the \(m\)th term of this new sequence without writing out all the preceding terms. Consider specifically the cases of \(m=1968\) and \(m=3968\).

How can we express the sum \(b_{1} + b_{2} + \ldots + b_{m}\) using the terms of the original sequence?



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