
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
Given that in triangle \( \triangle ABC \), \( AB = 2 \), \( AC = 3 \), \( \angle BAC = 60^\circ \), \(\overrightarrow{CD} = 2\overrightarrow{BE}\), \(\overrightarrow{AE} = x\overrightarrow{AD} + (1-x)\overrightarrow{AB}\), where \( x \in (0,1) \). Find the range of the projection of \( AE \) on \( AC \).

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