
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
If a finite sequence $a_1, a_2, a_3, \ldots, a_m$ (where $m$ is a positive integer) satisfies the conditions $a_1 = a_m$, $a_2 = a_{m-1}$, $\ldots$, $a_m = a_1$, i.e., $a_i = a_{m-i+1}$, we call it a "symmetric sequence". If $\{c_n\}$ is a 19-term "symmetric sequence", where $c_{10}, c_{11}, \ldots, c_{19}$ form a geometric sequence with the first term being 1 and the common ratio being 2, then $c_{19} = \_\_\_\_\_$, and $S_{19} = \_\_\_\_\_$.

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