
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 quadratic equation in $x$: $x^{2}-3x+2a+1=0$ has two distinct real roots.<br/>$(1)$ Find the range of real numbers for $a$;<br/>$(2)$ If $a$ is the largest integer that satisfies the condition, and the two roots of the quadratic equation $x^{2}-3x+2a+1=0$ are $x_{1}$ and $x_{2}$, find the value of $x_{1}^{2}x_{2}+x_{1}x_{2}^{2}$.

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