
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
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(xy + a) + f(xz + a) - f(x + a) f(yz + a) \ge k\]
for all real numbers $x,$ $y,$ and $z,$ where $a$ and $k$ are known real constants, and $k \ge 0$.

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