
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 vectors in the plane, and suppose a mapping $f$:

$V \rightarrow \mathbb{R}$ satisfies:

For any vectors $a = (x_1, y_1) \in V$, $b = (x_2, y_2) \in V$, 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 1, 2, and 3 have property $p$.

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