
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
As shown in the diagram, \( M \) is the midpoint of the median \( AD \) of \( \triangle ABC \). A line passing through point \( M \) intersects sides \( AB \) and \( AC \) at points \( P \) and \( Q \), respectively. Let \(\overrightarrow{AP} = x \overrightarrow{AB}\) and \(\overrightarrow{AQ} = y \overrightarrow{AC}\). Denote \( y = f(x) \).

1. Find the expression of the function \( y = f(x) \).
2. Let \( g(x) = x^3 + 3a^2x + 2a \) for \( x \in [0, 1] \). If for any \( x_1 \in \left[\frac{1}{3}, 1\right] \), there always exists \( x_2 \in [0, 1] \) such that \( f(x_1) = g(x_2) \) holds, determine the range of the real number \( a \).

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