
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  $a\in\mathbb{R}$  for which there exists a non-constant function  $f: (0,1]\rightarrow\mathbb{R}$  such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all  $x,y\in(0,1].$ 

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