
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
If the common difference $d$ of the arithmetic sequence $\{a_n\}$ is less than $0$, and $a_2 \cdot a_4 = 12$, $a_2 + a_4 = 8$.

$(1)$ Find the first term $a_1$ and the common difference $d$ of the sequence $\{a_n\}$;  
$(2)$ Calculate the sum of the first $10$ terms $S_{10}$ of the sequence $\{a_n\}$.

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