
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 $\mathbf{N}$ denote the set of natural numbers. The function $f: \mathbf{N} \rightarrow \mathbf{N}$ satisfies the conditions: $f(1)=1$, and for any natural number $n$,
$$
\left\{\begin{array}{l}
3 f(n) f(2 n+1)=f(2 n)(1+3 f(n)) \\
f(2 n)<6 f(n)
\end{array}\right.
$$

Find all solutions to the equation
$$
f(k)+f(l)=293, \quad k<l
$$

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