
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 $x \in \mathbb{R}$, the function $f(x)$ satisfies $f(x+1)= \sqrt {f(x)-[f(x)]^{2}}+ \frac {1}{2}$. Let $a_{n}=[f(n)]^{2}-f(n)$, and the sum of the first $15$ terms of the sequence $\{a_{n}\}$ is $- \frac {31}{16}$. Find $f(15)$.

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