
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
Determine all functions  $f : \mathbb{R} \to \mathbb{R}$  that satisfy the following two properties.
(i) The Riemann integral  $\int_a^b f(t) \mathrm dt$  exists for all real numbers  $a < b$ .
(ii) For every real number  $x$  and every integer  $n \ge 1$  we have
\[ f(x) = \frac{n}{2} \int_{x-\frac{1}{n}}^{x+\frac{1}{n}} f(t) \mathrm dt. \]

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