
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, given $\overrightarrow{a}=(3,-4)$, $\overrightarrow{b}=(2,-3)$.  
$(1)$ If $(\overrightarrow{a}-λ\overrightarrow{b})∥(3\overrightarrow{a}+\overrightarrow{b})$, find the value of the real number $\lambda$.  
$(2)$ If $\overrightarrow{c}=(0,4+\sqrt{3})$, $\overrightarrow{d}=(-1,3+\sqrt{3)$, find the dot product of $\overrightarrow{a}+\overrightarrow{c}$ and $\overrightarrow{b}+\overrightarrow{d}$.

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