
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 the function $f(x)=(x-1- \frac {a}{6})e^{x}+1$, where $e=2.718\ldots$ is the base of the natural logarithm, and the constant $a > 0$.  
(I) Find the number of zeros of the function $f(x)$ in the interval $(0,+\infty)$.  
(II) Does there exist infinitely many $a\in(1,4)$ such that $\ln a$ is a point of local maximum for the function $F(x)$, where the derivative of $F(x)$ is $F′(x)=(e^{x}-a)f(x)$? Explain your reasoning.

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