
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
Roger has exactly one of each of the first 30 states' new U.S. quarters. The quarters were released in the same order that the states joined the union. A graph shows the number of states that joined the union in each decade. What fraction of Roger's 30 coins represents states that joined the union during the decade 1800 through 1809? Express your answer as a common fraction.

(note: every space represents 2 states.)
[asy]size(200);
label("1780",(6,0),S);
label("1800",(12,0),S);
label("1820",(18,-12),S);
label("1840",(24,0),S);
label("1860",(30,-12),S);
label("1880",(36,0),S);
label("1900",(42,-12),S);
label("1950",(48,0),S);
label("to",(6,-4),S);
label("to",(12,-4),S);
label("to",(18,-16),S);
label("to",(24,-4),S);
label("to",(30,-16),S);
label("to",(36,-4),S);
label("to",(42,-16),S);
label("to",(48,-4),S);
label("1789",(6,-8),S);
label("1809",(12,-8),S);
label("1829",(18,-20),S);
label("1849",(24,-8),S);
label("1869",(30,-20),S);
label("1889",(36,-8),S);
label("1909",(42,-20),S);
label("1959",(48,-8),S);
draw((0,0)--(50,0));
draw((0,2)--(50,2));
draw((0,4)--(50,4));
draw((0,6)--(50,6));
draw((0,8)--(50,8));
draw((0,10)--(50,10));
draw((0,12)--(50,12));
draw((0,14)--(50,14));
draw((0,16)--(50,16));
draw((0,18)--(50,18));
fill((4,0)--(8,0)--(8,12)--(4,12)--cycle,gray(0.8));
fill((10,0)--(14,0)--(14,5)--(10,5)--cycle,gray(0.8));
fill((16,0)--(20,0)--(20,7)--(16,7)--cycle,gray(0.8));
fill((22,0)--(26,0)--(26,6)--(22,6)--cycle,gray(0.8));
fill((28,0)--(32,0)--(32,7)--(28,7)--cycle,gray(0.8));
fill((34,0)--(38,0)--(38,5)--(34,5)--cycle,gray(0.8));
fill((40,0)--(44,0)--(44,4)--(40,4)--cycle,gray(0.8));
[/asy]

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