
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 inequality $ax^{2}+x+c > 0$ has the solution set $\{x|1 < x < 3\}$.  
$(1)$ Find the values of $a$ and $c$;  
$(2)$ If the solution set of the inequality $ax^{2}+2x+4c > 0$ is $A$, and the solution set of the inequality $3ax+cm < 0$ is $B$, and $A \subset B$, find the range of the real number $m$.

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