专题6.3等比数列及其前n项和十一大题型（原卷版）_11.4-2_2026-11-04

      
      
专题6.3 等比数列及其前n 项和【十一大题型】

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 {𝑎𝑛}!9 n 1:[Â𝑎𝑛+1 = 2𝑎𝑛−2,𝑆2 = 10[
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 {𝑎𝑛}!9𝑛1:[Æ 𝑎1 = 1,𝑛𝑆𝑛+1−(𝑛+ 1)𝑆𝑛= 𝑛2 +𝑛.
(1)P{𝑎𝑛}!012”
(2)Â𝑏𝑛= (−1)𝑛𝑎𝑛+ [(−1)𝑛+ 1]2𝑛[P
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!
 {𝑐𝑛}Īī𝑐𝑛𝑐𝑛+2−𝑐2
𝑛+1 = 𝑘𝑐𝑛𝑐𝑛+1u𝑛∈𝑁*,𝑘»€

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5
8,𝑎2 =
15
16,3𝑎4 = 2𝑎5.
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(2)Ò𝑏𝑛=
𝑎𝑛,𝑛为奇数
𝑏𝑛−1 + 1,𝑛为偶数
 
[P
 {𝑏𝑛}!9𝑛1:𝑆𝑛.
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 {𝑎𝑛}!9𝑛1:»𝑆𝑛[
 {𝑏𝑛}!9𝑛1:»𝑇𝑛[4𝑆𝑛= 𝑎2
𝑛
+2𝑎𝑛+1[4𝑇𝑛= 𝑏2
𝑛+2𝑏𝑛+1
(1)P
 {𝑎𝑛},{𝑏𝑛}!012”
(2)Æ 
 {𝑐𝑛}Īī𝑐𝑛= 𝑏𝑛⋅
𝑎𝑛+1
𝑎𝑛𝑎𝑛+1[P
 {𝑐𝑛}!9𝑛1:𝐻𝑛¢
      
      
Ă2 8-3u2024·Ńńńģ·ħĬĭĮvÆ cã1
 {𝑎𝑛}3[𝑎3 = 4,𝑎2𝑎5 = 32[Ýln𝑎𝑛,ln𝑎𝑛+1,ln𝑎𝑛+2
E@
 .
(1)P
 {𝑎𝑛}!012”
(2)Â
 {𝑏𝑛}Īī𝑏𝑛= 𝑎𝑛+ (−1)𝑛log2𝑎𝑛+1[P
 {𝑏𝑛}!9𝑛1:𝑇𝑛.
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 -
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Z 9u2024·Ļļdžť·îħvÆ Å1» 1 !@
 {𝑎𝑛}Īī
𝑎1,𝑎2,𝑎3 +1E
 .
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 {𝑎𝑛}!012”
(2)Â
 {𝑏𝑛}Īī
𝑎1𝑏𝑛+ 𝑎2𝑏𝑛−1 +⋯+ 𝑎𝑛𝑏1 = 3𝑛−1[P
 {𝑏𝑛}!9𝑛1:𝑇𝑛.
Ă2 9-1u2024·ŷLJ·QvÆ {𝑎𝑛}z@
 [𝑎2 + 𝑎5 = 16,𝑎5−𝑎3 = 4¢
(1)P{𝑎𝑛}!012:
2𝑛−1
𝑖=2𝑛−1𝑎𝑖(𝑛∈N∗)¢
(2)Ò{𝑏𝑛}z
 [ÝþŚ^!𝑘∈N*[Ç2𝑘−1 ≤𝑛≤2𝑘−1È[¹𝑏𝑘< 𝑎𝑛< 𝑏𝑘+1[
uⅠvÇ𝑘≥2È[P.
2𝑘−1 < 𝑏𝑘< 2𝑘+1”
uⅡvP{𝑏𝑛}!012)9𝑛1:¢
Ă2 9-2u2024·pq·ħĬĭĮvÆ @
 {𝑎𝑛}!9 n 1:»𝑆𝑛[𝑎1 + 𝑎2 +3𝑎4 = 25[Ý𝑎3 +2[
𝑎4[𝑎5−2E
 ¢
      
      
(1)P
 {𝑎𝑛}!012”
(2)Ò𝑏𝑛= 𝑎𝑛⋅
3𝑎𝑛+1[P
 {𝑏𝑛}!9 n 1:𝑇𝑛¢
Ă2 9-3u2023·ŷLJLjĵ·îħvÒ{𝑎𝑛}z@
 [{𝑏𝑛}zË1­»ã
!
 ¢Ý𝑎1 = 𝑏1
= 1[𝑎3 + 𝑏2 = 7[2𝑎2−𝑏3 = 2[𝑛∈N∗
(1)P{𝑎𝑛}[{𝑏𝑛}!012”
(2)Ŧ𝑇𝑛»{𝑏𝑛}!9𝑛1:[P.
𝑇𝑛⋅𝑇𝑛+2 < 𝑇2
𝑛+1”
(3)Â𝑐𝑛=
(𝑎𝑛+ 1)⋅
𝑏𝑛,𝑛为奇数
3𝑏𝑛
𝑏𝑛−1
2
𝑏𝑛+2−1
2
,𝑛为偶数
 
[P
 {𝑐𝑛}!92𝑛1:𝑆2𝑛¢
 10  
 3!C2DEFGHI8
Z 10u2024·ĨńljŒ·îħvÆ 
 {𝑎𝑛}!9 n 1:»𝑆𝑛[Ý4𝑎1−3𝑆𝑛=
1
4𝑛−1¢
(1)P
 {𝑎𝑛}!012”
(2)Ò𝑏𝑛= 𝑛𝑎𝑛[Ý
 {𝑏𝑛}!9 n 1:»𝑇𝑛[Â∀𝑛∈𝑁∗­HC2𝑇𝑛≤
16
9 +3𝜆𝑎𝑛DEF[P𝜆!Ö7NJNj¢
Ă2 10-1u23-24 ïä·ţx·ř3vÆ 
 {𝑎𝑛}!9𝑛1:»𝑆𝑛[ÝĪī𝑆𝑛= 2𝑎𝑛−2.
 {𝑏𝑛}!9𝑛
1:»𝑇𝑛[ÝĪī𝑏1 = 1[
1
𝑏1𝑏2 +
1
𝑏2𝑏3 +⋯⋯+
1
𝑏𝑛𝑏𝑛+1 = 1−
1
𝑏𝑛+1(𝑛∈N*).
(1)P
 {𝑎𝑛},{𝑏𝑛}!012”
(2)Â𝑐𝑛= 𝑎𝑛𝑏𝑛[Ò
 {𝑐𝑛}!9𝑛1:»𝐻𝑛[ÝþŚ^!𝑛∈N*,𝐻𝑛−[𝑛−(−1)𝑛𝑚]𝑎𝑛+1 < 0DEF[P𝑚!Ö
      
      
7NJNj.
Ă2 10-2u2024·ţĶ·ïħvÆ {𝑎𝑛}zË1­»ã
!
 [
 {𝑏𝑛}Īī
𝑏𝑛= 2log2𝑎𝑛+1[
Ý𝑏1 = 1[𝑏4 = 7.
(1)P
 {𝑎𝑛},{𝑏𝑛}!012”
(2)ÂþŚ^!𝑛∈N*­H2𝜆𝑎𝑛≥𝑏𝑛−2[PY
𝜆!Ö7NJNj.
Ă2 10-3u2024·ŷLJnjǍ·‚ħvÆ 𝑆𝑛»
 {𝑎𝑛}!9 n 1:[ÝĪī𝑆𝑛= 2𝑎𝑛+𝑟[M3𝑟∈R[Ý
𝑟≠0¢
(1)P
 {𝑎𝑛}!012”
(2)Ò𝑏𝑛= (−1)𝑛+1𝑆𝑛
𝑟[ÂþŚ^!𝑛∈N∗[­H
2𝑛−1
𝑖=1 𝑏𝑖< 𝑚<
2𝑛
𝑖=1𝑏𝑖[PY
 m !Ö7NJNj¢
 11  -
 H!,JGKL8
Z 11u2024·pq·ħĬĭĮvƩ
[ʼnºǎ
.¬!,J§ä
ÂǏ
𝑎ǐ‹Ǐ
𝑚(𝑚≠0)ÃÜ!Ǒã
ǒzǏ
÷ǓHǔ
[ǕǖǗº𝑎»𝑚!Ž
[º𝑚»𝑎!Ʃ
.ÒãǏ
𝑎åH𝑘©ãƩ
[Ä»𝑎1[𝑎2,⋯,𝑎𝑘−1
,𝑎𝑘(𝑎1 < 𝑎2 < ⋯< 𝑎𝑘).
(1)Â𝑎= 8[P𝑘!7”
(2)Ç𝑘≥4È[Â{𝑎𝑘−𝑎𝑘−1}»
 [PãǏ
𝑎”
(3)Ŧ𝐴= 𝑎1𝑎2 + 𝑎2𝑎3 +⋯+ 𝑎𝑘−1𝑎𝑘[./
𝐴< 𝑎2.
      
      
Ă2 11-1u2024·ǘƲǙĦ·ħĬĭĮvÂŕŖ
 {𝑎𝑛}Īī
þ∀𝑛∈N*，𝑎2
𝑛+1−𝑎2
𝑛= 𝑝[M3𝑝»€
[
¹º
 {𝑎𝑛}»𝑃
 ¢
(1)‚©»𝑞!
 {𝑥𝑛}»“𝑃
 ”[P𝑞!7”
(2)Â𝑎1 = 1，𝑝= 2，{𝑦𝑛}zÅ1» 1[» 3 !
 [c𝑦𝑘-𝑦𝑘+1¾Ƈĝ¿À
 𝑎2
𝑛3!𝑘1â
E
 {𝑐𝑛}:𝑦1,𝑎2
1,𝑦2,𝑎2
2,𝑎2
3,𝑦3,𝑎2
4,𝑎2
5,𝑎2
6,𝑦4, ⋅⋅⋅⋅⋅⋅[P
 {𝑐𝑛}39 30 1!:𝑆30¢
(3)‚©“𝑃
 "{𝑎𝑛}Īī𝑎1 = 2,𝑎2 = 2 2,𝑎𝑛> 0[Ò
 
1
𝑎𝑛!9𝑛1:»𝑇𝑛¢zǚücãǏ
𝑚,𝑘[ÁC
2𝑇𝑛>
𝑚𝑛+ 𝑘−1þ‚Ǜ𝑛∈𝑁∗­EFǜÂüc[P—𝑚,𝑘!7”ÂCüc[Ņ/Ė¢
Ă2 11-2u2025·ǝǞưǟ·ħĬĭĮv,J
c‚©HŖ
 !«kàǠ1¾¿À±Ǡ1!:[E
!
 [Ǖǖǡ±Ƶ!Ǣƅº»ƥ
 !‚ĝ“:ǣŀ”[Z§

 1,2,3ƳVs‚ĝ“:ǣŀ”żÜū
 
1,3,2,5,3”sïĝ“:ǣŀ”żÜū
 1,4,3,5,2,7,5,8,3¢Ò
 𝑎,𝑏,𝑐ƳV𝑛ĝ“:ǣŀ”żÜū!
 !1
»
𝑃𝑛[ÃH1!:»𝑆𝑛¢
(1)Â𝑎= 2,𝑏= 3,𝑐= 4[P𝑃2,𝑆2”
(2)PC2𝑃𝑛≥2024!IǤ”
(3)zǚüc
 𝑎,𝑏,𝑐(𝑎,𝑏,𝑐∈𝑅)[ÁÜ
 {𝑆𝑛}»
 ǜœŅ/Ė¢
Ă2 11-3u2024·ĨĩĨĦ·ħĬĭĮvÂŕŖ1
 {𝑎𝑛}Īī𝑎𝑛+1 =
𝑎𝑛+ 𝑑，
𝑛
𝑡∉𝑁∗，
𝑞𝑎𝑛，
𝑛
𝑡∈𝑁∗，
 
u𝑑[𝑞[𝑡»€

[𝑡∈𝑁*Ý𝑡≥2v[¹º
 {𝑎𝑛}»“𝑀(𝑡)
 ”.
(1)Ò𝑑= 1[𝑞= 1[ÂÅ1» 1 !
 {𝑎𝑛}»“𝑀(3)
 ”[P𝑎2024”
      
      
(2)ÂÅ1» 1 !
 {𝑏𝑛}»“𝑀(𝑡)
 ”[P
 {𝑏𝑛}!012)9𝑛1:𝑆𝑛”
(3)Ò𝑑= 1[𝑞= 2[ÂÅ1» 1 !
 {𝑐𝑛}»“𝑀(5)
 ”[Ŧ
 {𝑐𝑛}!9𝑛1:»𝑇𝑛[PÃHĪī𝑇5𝑛= 5𝑛×
𝑐5𝑛−10𝑛!𝑛7.
‚G4
1¢u2024·ĴĩǥǦ·ïħvÆ 
 {𝑎𝑛},𝑎2＝4,𝑎10＝16,¹𝑎6 = uǧǧv
A¢8
B¢±8
C¢10
D¢±10
2¢u2024·Ńńģƫ·ħĬĭĮvÆ 
 {𝑎𝑛}!Ë1Ì»Ǩ
[ŦM9𝑛1:»𝑆𝑛[Â𝑆6−𝑆4 = −3,𝑎6𝑎7
𝑎8 = −
1
8[¹𝑎2 = u    v
A¢ǩ8
B¢ǩ16
C¢ǩ32
D¢ǩ48
3¢u2024·Ńńńģ·îħvÆ 𝑆𝑛z
 {𝑎𝑛}!9 n 1:[𝑎1 + 𝑎4 + 𝑎7 = 2[𝑎2 + 𝑎5 + 𝑎8 = 4[¹𝑆9 =
u    v
A¢12
B¢14
C¢16
D¢18
4¢u2024·Ŀń·ïħvÆ 
 {𝑎𝑛}!Å1𝑎1»€
Ý𝑎1 ≠
2
3[𝑎𝑛+1 +2𝑎𝑛= 4𝑛(𝑛∈𝑁*)[Â
 {𝑎𝑛}z¼É

 [¹𝑎1!Ö7NJNj»u    v
A¢ −2
3 , 2
3
B¢ −2
3 , 2
3 ∪
2
3 , 4
3
C¢ 0, 2
3
D¢ 0, 2
3 ∪
2
3 , 4
3
5¢u2023·ǪĦǫJ·ħĬĭĮvƤ9 1650 o!Ǭ)ǭǎǮǯǰDZĔDzH§ä8
“dzǴTdzǵŋǶ[
ƒsïǴűǷ[ËǴÃÜƇĝ9ǴǸǹT‚[8«ǴËÜŋǶǺǸǵǜ”cĔŔ83[9ëǴÜū!
ŋǶĠ$»u     v
A¢10×85
85+75ǵ
B¢10×86
85+75ǵ
C¢10×85
85−75ǵ
D¢10×86
85−75ǵ
6¢u2024·Ĩĩĩǻ·ħĬĭĮv@
 {𝑎𝑛}:
 {𝑏𝑛}­zË1»ãY
!ŕŖ
 [Ý𝑎1 = 𝑏1[
      
      
𝑎2 = 𝑏2[{𝑎𝑛}!9 n 1:»𝑆𝑛[{𝑏𝑛}!9 n 1:»𝑇𝑛[ä +Ǽãŝ!zu    v
A¢{𝑎𝑛}z¼É
 
B¢{𝑏𝑛}z¼É
 
C¢𝑆𝑛> 𝑇𝑛
D¢𝑆𝑛≤𝑇𝑛
7¢u2024·xyńǽ·ïħvÆ { 𝑎𝑛 }zŕŖ
 [M9𝑛1:»𝑆𝑛[ 𝑎 1  = 3 , 𝑆2 =
3
2¢ÂþŚ^ãǏ
𝑛[­H𝑆𝑛−(−1)𝑛⋅𝐴> 0[¹𝐴!Ö7NJNjzu    v
A¢( −3 ,1 )
B¢[ −2 ,1 )
C¢( −3 ,
3
2 )
D¢[ −2 ,
3
2 )
8¢u2024·ŃńǑǾ·ħĬĭĮvÒ
 {𝑎𝑛}!9𝑛1:»𝑇𝑛[9𝑛1á»𝐾𝑛[Â𝐾7 > 𝐾6 > 0[𝐾7 = 𝐾8 >
𝐾9 > 0[¹ä ġ?Cãŝ!zu    v
A¢𝑎8 = 1
B¢þŚ^ãǏ
𝑛[𝑎2
𝑛+ 𝑎2
𝑛+2 > 2𝑎2
𝑛+1
C¢𝐾10 > 𝐾6
D¢
 {𝑇2𝑛+2−𝑇2𝑛}‚,z
 
ïGǺ
9¢u2024·Ĩń·ħĬĭĮvÂ
 {𝑎𝑛} {𝑏𝑛}Īī𝑎𝑛+1 = 2𝑎𝑛+ 𝑏𝑛[𝑏𝑛+1 = 2𝑏𝑛+ 𝑎𝑛[Ý𝑎1 = 1[𝑏1 = 0[¹
ä ġ?3ãŝ!zu    v
A¢𝑎4 = 14
B¢𝑎𝑛+ 𝑏𝑛= 3𝑛−2
C¢
5
𝑖=1𝑏𝑖= 58
D¢Â{𝑎𝑛+ 𝜆𝑏𝑛}»
 [¹𝜆=± 1
10¢u2024·Ŀń·ħĬĭĮvÆ 
 {𝑎}!9𝑛1:»𝑆𝑛,𝑎1 = 1,𝑎𝑛+1 =
𝑎𝑛−1,𝑛= 2𝑘
2𝑎𝑛+ 1,𝑛= 2𝑘−1(𝑘∈N*)
 
[¹ä 
1ãŝ!zu    v
A¢𝑎4 = 5
B¢
 {𝑎2𝑘−1}z» 2 !
 
C¢𝑎𝑛= 2
𝑛+1
2 −1(𝑛= 2𝑘−1,𝑘∈N*)
D¢𝑆𝑛≤50!6Ǐ
𝑛!7» 8
11¢u2024·ţĶǿť·îħvÆ {𝑎𝑛}z
 [𝑆𝑛zM9 n 1:[Īī𝑎3 = 2𝑎1 + 𝑎2[¹ä Ņøãŝ
!Hu    v
A¢Â{𝑎𝑛}zã1
 [¹{𝑎𝑛}z45¼É
 
B¢𝑆𝑛,𝑆2𝑛−𝑆𝑛,𝑆3𝑛−𝑆2𝑛‚,z
 
      
      
C¢Âüc𝑀> 0[Á|𝑎𝑛| ≤𝑀þ∀𝑛∈N+­EF[¹{|𝑎𝑛|}z@
 
D¢Â𝑎𝑛> 0[Ý𝑎1 =
1
100[𝑇𝑛= 𝑎1 ⋅𝑎2⋯𝑎𝑛[¹𝑛= 7È𝑇𝑛Ö6Ş7
îGŽ
12¢u2024·ĻļȀģ·îħv
 {𝑎𝑛}3[«1Ì»ã
[Ý𝑎4 ⋅𝑎5 = 81[¹log3𝑎1 + log3𝑎8 =      
.
13¢u2024·ţxȁť·ïħvÆ @
 {𝑎𝑛}:
 {𝑏𝑛}Īī𝑎1 = 5[𝑏1 = 2[𝑎2 = 2𝑏2 +1[𝑎3 = 𝑏3
+5.
 {𝑎𝑛}:{𝑏𝑛}3!ÃH1TȂâEǤB𝐴、𝐵[ȃǤB𝐴∪𝐵3!ÃHƤȄ׃ŞūƇĝÛ âE‚
©
 {𝑐𝑛}[
 {𝑐𝑛}!9𝑛1:»𝑆𝑛[¹𝑆20 =      
.
14¢u2024·xy0Ħ·îħvÂ
 {𝑏𝑛}G{𝑐𝑛}Ì»şŠÉ
 [ÝþŚ^ãǏ
 n[­ücãǏ
 m[ÁÜ
𝑏𝑚∈[𝑐𝑛,𝑐𝑛+1][¹º
 {𝑏𝑛}»
 {𝑐𝑛}!“M 
 ”¢Æ 
 {𝑎𝑛}!9 n 1:»𝑆𝑛[¹ä ġ?3ãŝ!
z      
¢
①üc@
 {𝑎𝑛}[ÁÜ{𝑎𝑛}z{𝑆𝑛}!“M 
 ”
②üc
 {𝑎𝑛}[ÁÜ{𝑎𝑛}z{𝑆𝑛}!“M 
 ”
③üc@
 {𝑎𝑛}[ÁÜ{𝑆𝑛}z{𝑎𝑛}!“M 
 ”
④üc
 {𝑎𝑛}[ÁÜ{𝑆𝑛}z{𝑎𝑛}!“M 
 ”
ĻGI˜
15¢u2024·ȅİ·ħĬĭĮvÆ 
 {𝑎𝑛}!9𝑛1:»𝑆𝑛[Ý4𝑆𝑛= 5𝑎𝑛−2¢
(1)./
{𝑎𝑛}z
 [iPM012”
(2)Ò𝑏𝑛= (−1)𝑛⋅log5
𝑎2𝑛+2
2 [P
 {𝑏𝑛}!9 100 1:𝑇100¢
16¢u2024·ŷLJķń·ħĬĭĮvÆ Ȇ𝐴03ȇH 3 ƙȈ[Ȇ𝐵03ȇH 1 ƙȈ.hȃȆ𝐴03!Ȉ!
2
3:Ȇ𝐵03
!Ȉ!
1
3ȉÀȆ𝐴13[ƆȃȆ𝐴0-Ȇ𝐵03Ȋǔ!ȈȉÀȆ𝐵13”ȋżȃȆ𝐴13!Ȉ!
2
3:Ȇ𝐵13!Ȉ!
1
3ȉÀ
Ȇ𝐴23[ƆȃȆ𝐴1-Ȇ𝐵13Ȋǔ!ȈȉÀȆB23”§ėȌßǢƅä™.
(1)PǢƅ 1 ĝżȆ𝐵13!Ȉ$”
(2)PǢƅ𝑛ĝżȆ𝐵𝑛3!Ȉ$”
      
      
(3)ƊǸǢƅǺǸĝ[Ȇ𝐴𝑛(𝑛∈𝑁∗)3!Ȉ$-Ȇ𝐵𝑛(𝑛∈𝑁∗)3!Ȉ$@Ş
1
64ƙǜuƪ
Ą
lg
2 ≈0.3010[lg3 ≈0.4771v
17¢u2024·ĻļǮť·îħvÆ {𝑎𝑛}z@
 [{𝑏𝑛}z
 [Ý{𝑏𝑛}!9𝑛1:»𝑆𝑛[2𝑎1 = 𝑏1 = 2[
𝑎5 = 5(𝑎4−𝑎3)[c①𝑏5 = 4(𝑏4−𝑏3)[②𝑏𝑛+1 = 𝑆𝑛+2±Ǡ©Ćć3ŚM3‚©[ŸEäƏ8!I˜.
(1)P
 {𝑎𝑛}:{𝑏𝑛}!012”
(2)Ò
 𝑎𝑛
𝑏𝑛!9𝑛1:»𝑇𝑛[P𝑇𝑛.
18¢u2024·ŷLJ·ïħvÒ{𝑎𝑛}z@
 [M9𝑛1:𝑆𝑛[{𝑏𝑛}z
 [Ý𝑎1 = 𝑏1 = 3[𝑎4 = 𝑏2[𝑆3
= 15.
(1)P{𝑎𝑛}-{𝑏𝑛}!012”
(2)Ò𝑐𝑛=
𝑎𝑛𝑏𝑛,𝑛为奇数
(3−4𝑛)𝑏𝑛
𝑎𝑛−1·𝑎𝑛+1 ,𝑛为偶数
 
[P
 {𝑐𝑛}!92𝑛1:𝑇2𝑛”
(3)ÂþŚ^!𝑛∈N*C2𝑛(𝑎𝑛+ 1)−𝜆(𝑎𝑛−1)(𝑛+ 2)−12 < 0DEF[PY
𝜆!Ö7NJNj.
19¢u2024·•ŧűĦ·ħĬĭĮvÒHŖ
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