
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: $$f(x+1)= \sqrt {f(x)-f^{2}(x)}+ \frac {1}{2}$$, and $$a_{n}=f^{2}(n)-f(n)$$. The sum of the first 15 terms of the sequence $\{a_n\}$ is $- \frac {31}{16}$. The sequence $\{c_n\}$ satisfies $$c_{n}+c_{n+1}=[f(2019)]^{n}$$. Find $c_1$ if the limit of the sum of the first $n$ terms of the sequence $\{c_n\}$ exists.

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