
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
Let  $f:\mathbb{R}^+\to \mathbb{R}^+$  be a function such that for all  $x,y \in \mathbb{R}+,\, f(x)f(y)=f(xy)+f\left(\frac{x}{y}\right)$ , where  $\mathbb{R}^+$  represents the positive real numbers. Given that  $f(2)=3$ , compute the last two digits of  $f\left(2^{2^{2020}}\right)$ .

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