
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 polynomials P of two variables so that:

**a.)** for any real numbers  $t,x,y$  we have  $P(tx,ty) = t^n P(x,y)$  where  $n$  is a positive integer, the same for all  $t,x,y;$ 

**b.)** for any real numbers  $a,b,c$  we have  $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$ 

**c.)**  $P(1,0) =1.$ 

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