
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 functions $f_1(x)=x^2+x$, $f_2(x)=2x^2-x$, $f_3(x)=x^2+x$, $g_1(x)=x-2$, $g_2(x)=2x$, and $g_3(x)=x+2$, determine the tuple $(F_1,F_2,F_3)$ by evaluating whether $h(x)=x$ can be obtained from $f_i$ and $g_i$ through only addition, subtraction, and multiplication, where $i\in\{1,2,3\}$.

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