
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 $f(x)$ satisfies $f(1)=a$, and $f(n+1)= \begin{cases} \frac {f(n)-1}{f(n)},f(n) > 1 \\ 2f(n),f(n)\leqslant 1\\ \end{cases}$, if for any $n \in \mathbb{N}^{*}$, $f(n+3)=f(n)$ always holds, then the possible number of values of $a$ in $(0,1]$ is $\_\_\_\_\_\_$.

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