
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's denote by \( x \) the number of oranges the father had before distributing them among his sons. He gives the first son a number of oranges equal to \( \frac{x}{2}+\frac{1}{2} \); he is left with \( x-\left(\frac{x}{2}+\frac{1}{2}\right) = \frac{x}{2}-\frac{1}{2} \). Then he gives the second son half of the remainder, or \( \frac{1}{2}\left(\frac{x}{2}-\frac{1}{2}\right) \), increased by \( \frac{1}{2} \); this is \( \frac{1}{2}\left(\frac{x}{2}-\frac{1}{2}\right)+\frac{1}{2}=\frac{x}{4}+\frac{1}{4} \). Then he is left with \( \frac{x}{2}-\frac{1}{2}-\left(\frac{x}{4}+\frac{1}{4}\right)= \frac{x}{4}-\frac{3}{4} \). Finally, he gives the third son \( \frac{1}{2}\left(\frac{x}{4}-\frac{3}{4}\right) + \frac{1}{2} = \frac{x}{8}+\frac{1}{8} \). Since he has nothing left, we must have \( \frac{x}{8}+\frac{1}{8} = \frac{x}{4}-\frac{3}{4} \), from which \( x=7 \).

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