
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
In $\triangle ABC$, the lengths of sides $BC$, $AC$, and $AB$ are $a$, $b$, and $c$ respectively, with $a=4$. Point $E$ is the midpoint of side $BC$.
(1) If $\overrightarrow {AB}\cdot \overrightarrow {AC} = 1$, find the length of the median $AE$ from $A$ to side $BC$;
(2) If the area of $\triangle ABC$ is $3 \sqrt {2}$, find the minimum value of $\overrightarrow {AB}\cdot \overrightarrow {AC}$.

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