
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  $u_n$  be defined by  $u_0=0$  and  $u_{2n}=u_n$ ,  $u_{2n+1}=1-u_n$  for each  $n\in\mathbb N_0$ .
(a) Calculate  $u_{1990}$ .
(b) Find the number of indices  $n\le1990$  for which  $u_n=0$ .
(c) Let  $p$  be a natural number and  $N=(2^p-1)^2$ . Find  $u_N$ .

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