
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 a fixed point $A(12, 0)$, let $M$ be a moving point on the curve $$\begin{cases} x=6+2\cos\theta \\ y=2\sin\theta \end{cases}.$$ 
(1) If point $P$ satisfies the condition $\overrightarrow{AP} = 2\overrightarrow{AM}$, find the equation of the trajectory $C$ of the moving point $P$;  
(2) If line $l: y = -x + a$ intersects the curve $C$ at two distinct points $E$ and $F$, with $O$ being the origin and $\overrightarrow{OE} \cdot \overrightarrow{OF} = 12$, find the cosine of $\angle EOF$ and the value of the real number $a$.

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