
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

Investigate the following systems of equations:

a)
\[
\left\{
\begin{aligned}
2x + 3y & = 5 \\
x - y & = 2 \\
x + 4y & = a
\end{aligned}
\right.
\]

b)
\[
\left\{
\begin{aligned}
x + ay & = 1 \\
2x + 4y & = 2 \\
bx + 4y & = 2
\end{aligned}
\right.
\]

c)
\[
\left\{
\begin{aligned}
ax + by & = a \\
(a - 2)x + y & = 3 \\
x + y & = 1
\end{aligned}
\right.
\]

d)
\[
\left\{
\begin{array}{l}
x + ay + a^2z = a^3 \\
x + by + b^2z = b^3 \\
x + cy + c^2z = c^3
\end{array}
\right.
\]

e)
\[
\left\{
\begin{aligned}
x + y + z & = 1 \\
ax + by + cz & = d \\
a^2x + b^2y + c^2z & = d^2
\end{aligned}
\right.
\]

f)
\[
\left\{
\begin{aligned}
ax + by + cz & = a + b + c \\
bx + cy + az & = a + b + c \\
cx + ay + bz & = a + b + c
\end{aligned}
\right.
\]

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