
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
Among the following four propositions, the number of correct propositions is ______.
$(1)$ The negation of the proposition "If $f(x)$ is a periodic function, then $f(x)$ is a trigonometric function" is "If $f(x)$ is a periodic function, then $f(x)$ is not a trigonometric function";
$(2)$ The negation of the proposition "There exists $x \in \mathbb{R}$ such that $x^{2}-x > 0$" is "For every $x \in \mathbb{R}$, $x^{2}-x < 0$";
$(3)$ In $\triangle ABC$, "$\sin A > \sin B$" is both a necessary and sufficient condition for "$A > B$";
$(4)$ If a function $f(x)$ has a zero in the interval $(2015,2017)$, then it must hold that $f(2015) \cdot f(2017) < 0$;
$(5)$ The graph of the function $y = \ln x$ contains two points such that the tangents to the graph at these two points are perpendicular to each other.

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