
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
Define the length of the intervals \((a,b)\), \([a,b)\), \((a,b]\), \([a,b]\) as \(d=b-a\). The length of the union of multiple intervals is the sum of the lengths of each interval. For example, the length \(d\) of the union \((1,2) \cup [3,5)\) is \(d=(2-1)+(5-3)=3\). Let \(\lfloor x \rfloor\) denote the greatest integer less than or equal to \(x\), and let \(\{x\} = x - \lfloor x \rfloor\), where \(x \in \mathbb{R}\). Suppose \(f(x)=\lfloor x \rfloor \cdot \{x\}\) and \(g(x)=x-1\). If for \(0 \leqslant x \leqslant k\), the solution set of the inequality \(f(x) < g(x)\) has a length of 5, then the value of \(k\) is ______.

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