
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 $V$ be the set of all planar vectors. If the mapping $f$:  
$V \rightarrow \mathbb{R}$ satisfies:  
For any vector $a = (x_1, y_1) \in V$, $b = (x_2, y_2) \in V$, and any $\lambda \in \mathbb{R}$, it holds that $f[\lambda a + (1 - \lambda) b] = \lambda f(a) + (1 - \lambda) f(b)$, then the mapping $f$ is said to have property $p$.  
Now, consider the following mappings:  
① $f_1: V \rightarrow \mathbb{R}$, $f_1(m) = x - y$, where $m = (x, y) \in V$;  
② $f_2: V \rightarrow \mathbb{R}$, $f_2(m) = x^2 + y$, where $m = (x, y) \in V$;  
③ $f_3: V \rightarrow \mathbb{R}$, $f_3(m) = x + y + 1$, where $m = (x, y) \in V$.  
Analyze whether mappings ①②③ have property $p$.

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