
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 $y=f(x)$, if for any $x \in \mathbb{R}$, the equation $f(2x)=2f(x)$ always holds, then the function $y=f(x)$ is said to have property $P$.

$(1)$ If the function $y=f(x)$ has property $P$, and $f(4)=8$, then $f(1)=$\_\_\_\_\_\_\_\_;

$(2)$ If the function $y=f(x)$ has property $P$, and its expression on $(1,2]$ is $y=\cos x$, then $y=f(x)$ has exactly \_\_\_\_\_\_\_\_ zero(s) on $(1,8]$.

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