
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 $y=f(x)$ is defined and increasing on $(0,+\infty)$, and satisfies $f(xy)=f(x)+f(y)$, $f(4)=1$:
$(1)$ Find the value of $f(1)$;
$(2)$ If there exists a real number $m$ such that $f(m)=2$, find the value of $m$;
$(3)$ Find the range of $x$ if $f(x^{2}-4x-5) < 2$.

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