
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 $\rho^{2}= \dfrac {36}{4\cos ^{2}\theta +9\sin ^{2}\theta }$.
$(1)$ If the pole is taken as the origin of the Cartesian coordinate system, with the polar axis as the positive direction of the $x$-axis, establish the Cartesian coordinate system and find the equation of curve $C$ in Cartesian coordinates;
$(2)$ Let $P(x,y)$ be a moving point on curve $C$, find the maximum value of $3x+4y$.

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