
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 the Cartesian coordinate system, the parametric equation of curve $C_1$ is: 
$$
\begin{cases} 
x=4\cos \theta \\
y=3\sin \theta 
\end{cases}
(\theta \text{ is the parameter})
$$
With the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, the polar coordinate equation of curve $C_2$ is $\rho\sin (\theta+ \frac {\pi}{4})= \frac {5 \sqrt {2}}{2}$.  
$(1)$ Find the Cartesian coordinate equation of curve $C_2$;  
$(2)$ Given a point $M$ on curve $C_1$, find the range of the distance $d$ from point $M$ to curve $C_2$.

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