
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 $y=f(x)$, $x\in D$, if there exists a constant $C$ such that for all $x\_1\in D$, there exists a unique $x\_2\in D$ that satisfies $\sqrt{f(x\_1)f(x\_2)}=C$, then the constant $C$ is referred to as the "geometric mean multiple" of the function $f(x)$ on $D$. For the function $f(x)=2^{-x}$, $x\in[1,3]$, find the "geometric mean multiple" of $f(x)$ on $[1,3]$.

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