
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
Determine all real numbers  $a$  for which the system of equations
\begin{align*}
3x^2+2y^2+2z^2&=a
4x^2+4y^2+5z^2&=1-a
\end{align*}
has at least one solution  $(x,y,z)$  in the real numbers.

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