
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 $f(x)$, where $f(x)$ is an odd function defined on $\mathbb{R}$ and given by
$$
f(x) =
\begin{cases}
2^{x}, & \text{when $0 < x \leqslant 1$} \\
\frac {1}{2}f(x-1), & \text{when $x > 1$}
\end{cases}
$$
find the sum of all real roots of the equation $f(x) = \frac{1}{x}$ on the interval $[-3, 5]$.

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