
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 sequence \(a(n), n=1,2,3, \ldots\), be generated as follows: \(a(1)=0\), and for \(n>1\),
\[a(n)=a(\left\lfloor \frac{n}{2} \right\rfloor)+(-1)^{n+1} \]
(Here \(\left\lfloor t \right\rfloor\) is the greatest integer less than or equal to \(t\)).

(a) Determine the maximum and minimum values of \(a(n)\) over \(n \leq 1996\) and find all \(n \leq 1996\) for which these extreme values are attained.

(b) How many terms \(a(n)\), \(n \leq 1996\), are equal to 0?

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