
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 any pair of real numbers \(a\) and \(b\), consider the sequence of numbers \( p_{\mathrm{n}} = [2\{a n + b\}] \). Any \( k \) consecutive terms of this sequence are 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 \( b \) when \( k = 4 \); and when \( k = 5 \)? Note: \([c]\) denotes the integer part, and \(\{c\}\) denotes the fractional part of the number \(c\).

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