
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 generating functions for the sequence of Fibonacci polynomials $F(x, z) = F_{0}(x) + F_{1}(x) z + F_{2}(x) z^{2} + \ldots + F_{n}(x) z^{n} + \ldots$ 

and the sequence of Lucas polynomials $L(x, z) = L_{0}(x) + L_{1}(x) z + L_{2}(x) z^{2} + \ldots + L_{n}(x) z^{n} + \ldots$

Definitions of Fibonacci and Lucas polynomials can be found in a reference book.

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