
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
Define a function $f(x)$ on the interval $(0, +\infty)$ such that $f(x)+f(y)=f(xy)$, and $f(x) < 0$ for $x>1$. If the inequality $f(\sqrt{x^{2}+y^{2}})\leq f(\sqrt{xy})+f(a)$ holds true for any $x, y \in (0, +\infty)$, then the range of the real number $a$ is __________.

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