
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 the positive numbers  $a{}$  for which the following statement true: for any function  $f:[0,1]\to\mathbb{R}$  which is continuous at each point of this interval and for which  $f(0)=f(1)=0$ , the equation  $f(x+a)-f(x)=0$  has at least one solution.

*Proposed by I. Yaglom*

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