
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 the polar equation of curve $C$ is $\rho=4\cos \theta$. Taking the pole as the origin of the Cartesian coordinate system and the polar axis as the positive half-axis of $x$, establish a Cartesian coordinate system. The parametric equation of line $l$ is: 
$$
\begin{cases} 
x=m+ \dfrac { \sqrt {2}}{2}t \\
y= \dfrac { \sqrt {2}}{2}t
\end{cases}
(t \text{ is the parameter}).
$$
$(1)$ If line $l$ intersects curve $C$ at points $A$ and $B$, and $|AB|= \sqrt {14}$, find the value of the real number $m$.  
$(2)$ Let $M(x,y)$ be any point on curve $C$, find the range of values for $x+2y$.

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