
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
Fix positive integers  $k$  and  $n$ .Derive a simple expression involving Fibonacci numbers for the number of sequences  $(T_1,T_2,\ldots,T_k)$  of subsets  $T_i$  of  $[n]$  such that  $T_1\subseteq T_2\supseteq T_3\subseteq T_4\supseteq\ldots$ .

<span style="color:#008000">Moderator says: and the original source for this one is Richard Stanley, *Enumerative Combinatorics* vol. 1 (1st edition), exercise 1.15.</span>

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