
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
Find all real  $ a$ , such that there exist a function  $ f: \mathbb{R}\rightarrow\mathbb{R}$  satisfying the following inequality:
\[ x\plus{}af(y)\leq y\plus{}f(f(x))
\]
for all  $ x,y\in\mathbb{R}$ 

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