
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
For each pair of real numbers \(a\) and \(b\), consider the sequence of numbers \(p_n = \lfloor 2\{an + b\} \rfloor\). Any \(k\) consecutive terms of this sequence will be called a word. Is it true that any ordered set of zeros and ones of length \(k\) will be a word of the sequence, defined by some \(a\) and \(b\), for \(k = 4\); and for \(k = 5\)?

Note: \(\lfloor c \rfloor\) is the integer part of \(c\), and \(\{c\}\) is the fractional part of \(c\).

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