
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 equation in terms of $x$: $\left(a-5\right)x^{2}-4x-1=0$,<br/>$(1)$ If the equation has real roots, find the range of values for $a$;<br/>$(2)$ Does there exist a real number $a$ such that the two roots $x_{1}$ and $x_{2}$ of the equation satisfy $x_{1}+x_{2}+x_{1}x_{2}=3$? If yes, find the value of $a$; if not, explain the reason.

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