
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  $a_n$  be the number of sets  $S$  of positive integers for which
\[ \sum_{k\in S}F_k=n,\]
where the Fibonacci sequence  $(F_k)_{k\ge 1}$  satisfies  $F_{k+2}=F_{k+1}+F_k$  and begins  $F_1=1$ ,  $F_2=1$ ,  $F_3=2$ ,  $F_4=3$ . Find the largest number  $n$  such that  $a_n=2020$ .

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