
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 that the arithmetic sequence $\left\{ a_n \right\}$ is a decreasing sequence, and $a_1 + a_3 = -2$, $a_1a_3 = -3$.

$(1)$ Find the general formula for the sequence $\left\{ a_n \right\}$;   $(2)$ If the sum of the first $k$ terms of the sequence $\left\{ a_n \right\}$, $S_k = -35$, find the value of $k$.

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