
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
The function $f(x)$ is determined by a mapping $f$ from the vector set $\overrightarrow{A}$ to $\overrightarrow{A}$, and $f(x) = x - 2(x \cdot \overrightarrow{a})\overrightarrow{a}$. If there exists a non-zero constant vector $\overrightarrow{a}$ such that $f[f(x)] = f(x)$ always holds,  
(1) find $|\overrightarrow{a}|$;  
(2) let $\overrightarrow{AB} = \overrightarrow{a}$, $\overrightarrow{A} = (1, -2)$, if point P divides $\overrightarrow{AB}$ in the ratio of $-\frac{1}{3}$, find the equation of the curve on which point P lies.

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