
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
Let the continuous function \( f(x) \) be defined everywhere except at 0 and satisfy
$$
\frac{1}{f(x+y)} \leqslant \frac{f(x)+f(y)}{f(x) f(y)},
$$
with \( f(x) \neq 0 \) for \( x \neq 0 \) and \( \lim_{x \to \infty} f(x) = \infty \). Then \( f(x) = \frac{a}{x} \), where \( a \neq 0 \).

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