
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 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 $(1, 2) \cup [3, 5)$ is $(2-1) + (5-3) = 3$. Let $[x]$ denote the greatest integer not exceeding $x$, and let $\{x\} = x - [x]$, where $x \in \mathbb{R}$. Suppose $f(x) = [x]\cdot\{x\}$, $g(x) = x - 1$. When $0 \leq x \leq k$, the length of the solution set interval for the inequality $f(x) < g(x)$ is 5, then the value of $k$ is.

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