
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
Given that the function $f(x)$ defined on $[0,+\infty)$ satisfies $f(x+1)=2f(x)$, and when $x\in[0,1)$, $f(x)=-x^{2}+x$. Let $a_{n}$ denote the maximum value of $f(x)$ on $[n-1,n)$, where $n\in\mathbb{N}^{*}$. Determine the value of $a_{4}$.

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