
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 $x\in R$. Suppose $f(x)=[x]g\{x\}$, $g(x)=x-1$, when $0\leqslant x\leqslant k$, the length of the solution set interval of the inequality $f(x) < g(x)$ is $10$, then $k=$ \_\_\_\_\_\_.

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